Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/free-books/57/sergeev.pdf Äàòà èçìåíåíèÿ: Mon May 19 17:38:03 2008 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 11:53:51 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: comet

. .

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. . - . . -- .: , .-- . ISBN - , «» . , - . -- «». , -- , , .

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C



.

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ISBN

--

-

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© . ., © , .

.

. . . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . - . ( - (- ) . . . . . . . . . . . . . (A- ) . . . . . . . . . . . . . . . . . . . . (A- ) . . . . . . (- ) . . . . . . . . . . . . . . . . . (A- ) . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . - . (- ) . - . (- ) - . Le Bagatelle (- ) . . . . . . .

... ... ... ... ... ... ... ... ... ... ... ...


. . . . . . . . . . . . . . . . . . . -- (- ) . . . -- . (- ) . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . (- ) (- ) . . . . . . . . . . . (- ) . . . . . . . . . . . . . . (- ) . . . (- ) . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


, , - . , ( ) , -- . , ( ), ( ) ( ). . - . . [ ] [ ], . , , -- . . . . , . . . , . . , , , « ». , . , , . . , . . . , , . -- «». , -- , , . -- -- -- , «» .

. . . . . . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . (A- ) . . . . . . . . (A- ) . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . (- ) . ( . . (- ) . .

... ... ... ... ... ... ... ... ... ... - ) (... . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . ) .. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . . . (- ) (- ) . . . . . . . . . . . . . . . . . . --. (- ) . . . . . . . . . . . . . . . . . . . . . . . . Calculations (- ) . . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . (- ) . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .


. , - , . , , : . ( . ), . ( . ), . ( . ), . ( . ), . ( - . ), . . ( - ), . ( - . ), . (- . ). . (University of Chicago), . . (Amsterdam University), . (University of New York) . (). : . . , . . . . , , . , , . . [ ], . [ ] . . [ ], - , « » . . . , , , , .


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( .) , -- -- , , , . . , : -- ; -- ; [i ] -- , . . a + bi , a b -- , i 2 = -1; [ x ] -- , . . an x n + a , ; [[ x ]] -- , . . . a 0 + a 1 x + a 2 x 2 + ... + a n x n + ... ( a k ),
n -1

x

n -1

+ ... + a1 x + a

0

( a k ),

. : () (1 + x )(1 + x 2 )(1 + x 4 )(1 + x 8 )(1 + x 16 ); () (1 + x + x 2 + x 3 + ... + x 9 )2 ; () ( x 4 - 7 x 3 + 8 x 2 + 28 x - 48)/( x 2 - 4); () ( x 56 + x 55 + x 54 + ... + x 2 + x + 1)/( x 18 + x 17 + ... + x + 1). . P ( x ) Q ( x ) , . , P ( x )Q ( x ) . . : () (1 + i )(1 - i ) + 1; () (2 + 3i )(-3 - i ) + (-2 + 2i )(5 - 3i ); () 20i /(1 - 2i ); () (11i + 16)/(3 - 2i ); () 5 + 12i . . ( x , y ) x + iy = (2 + i )(a + bi ), -2 a, b 4.


() (1 - () (1 - () (1 + ( ) x /( 1 () (2 +

*. , ? M -- . , a M b M (b = 0), M . c, a = bc. : a . b. . . 111...1
1000

. : () 1 - 2 x + 3 x 2 - 4 x 3 + ...; () 1 + x .

() 1/(1 - x - x 2 ).

2 3 4 () (2 + 3 x + 3 x 2 + 3 x 3 + 3 x 4 + ...) 1 - 3 x + 3 x - 3 x + 3 x - ... ;

. : x )(1 + x + x 2 + x 3 + x 4 + ...); x + x 2 - x 3 + ... )(1 + x + x 2 + x 3 + ... ); x + x 2 + ... )2 ; + 4 x 2 ); x - x 2 )(1 - 2 x + 3 x 2 - 4 x 3 + ...);
2

- . - .
( .) . () :


4

8

16

(1 + x 2 ) = (1 + x )(1 + x 2 )(1 + x 4 )(1 + x 8 )...
k =0

k

() , 1, 2, 4, 8, 16, ... ( ) , . () , (, 5710 = 1110012 ), . . : (1 + x + x 2 + ... + x 9 )(1 + x 10 + x 20 + ... + x 90 )(1 + x
100

+x

200

+...+ x

900

)...

() ; () 11 111; () 111 111? . () x 1000 + x 999 + ... + x + 1 x 6 + x 5 + ... + x + 1 (); () x 4 + x - 2 x + 2 (); () 57 x - 2 (); () 57 x - 2 (); () 12 + 3i 2 + i ; () 6 + 17i 4i - 3?

. 13. *. () , 1+ x + 1
x

1 + x 3 + 13
x

1 + x 9 + 19
x x

1+ x

27

+1 2
x

7

...=
x x

= 1 + x + 1 + x 2 + 12 + x 3 + 13 + x 4 + 14 + ...
x

( ). () , 1, 3, 9, 27, ... ( ) , , , . . :
k =0

(1 - x 2 ) = (1 - x )(1 - x 2 )(1 - x 4 )(1 - x 8 )...

k

. a0 , a1 , a2 , a3 ... a0 + a1 x + a2 x 2 + a3 x 3 + ...


. fn f0 = 0, f1 = 1, fn = fn-1 + fn-2 n 2. , x . 2 . ,
P(x ) , P , Q -- . (Q(x ) 1- x - x

- . ( .)
, , ? . .
k . Cn k - n .
2 , C4 = 6, {1, 2, 3, 4} 6 :

, .) . , () 1, 3, 9, 27...; () 1, 3, 5, 7, 9... . *. , . . An x + 2 y + + 5z + 10t = n . , A0 = 1, A1 = 1, A2 = 2, A3 = 2, A4 = 3, A12 = 15. , 1 An . (1 - x )(1 - x 2)(1 - x 5 )(1 - x 10 ) . 100 1 2 5 10 ? . , (1 + x )(1 + x 2 )(1 + x 3 )(1 + x 4 )... = 1 . (1 - x )(1 - x 3 )(1 - x 5 )(1 - x 7 )...

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.

. , , . , 6 6, 1 + 5, 2 + 4, 1 + 2 + 3 1 + 5, 3 + 3, 1 + 1 + 1 + 3, 1 + 1 + 1 + 1 + 1 + 1, -- , .

k - M k - M . . () 3- 3 {1, 2, 3, 4, 5, 6} C6 . k n () , Cn = Cn -k . . «» . ? . () , n k Cn (1 + x )n . () ( .) ,
1 (a + b)n = an + Cn a n -1 2 b + Cn a n -2 2 k b + ... + Cn a n- k k

b +...
n -1

n n k Cn ak b k =0 n- k

...+C

n -1 n

ab

+b =

.

(, (1 + x )m+n = (1 + x )m (1 + x )n , (, « , ...»).

. , C
k +1 n +1

=C

k +1 n

k + Cn .

Isaak Newton ( -- ) -- , , ( ) .


. . , T . .

-

T

*. fm,n (0, 0) (m, n) â , (1, 0), (0, 1) (1, 1). fm,n x m y n , m, n 0 1 .
1 - x - y - xy

. () . () , k - n- k Cn .

.

. , , -- . , ? ( .) *. , ,


1 .
1 - y - xy

n , k =0

k Cn x k y n

. : 0 1 n () Cn + Cn + ... + Cn ; 1 3 0 2 () Cn + Cn + ... Cn + Cn + ...;
0s 1s () Cm Cn + Cm Cn-1 + ... ( ); 0 2 4 () Cn - Cn + Cn - ... 1 1 . 1 (. . ). 1 n- 1 ?

. A B, (. . )? (4, 2)? . ( ), C
02 n

1 2 3 4 15

1 31 641 10 10 5 1

.
B A

+C

12 n

+...+ C

n2 n

n = C2n .

.

Blaise Pascal (

--

) -- , .


A- .
( .)
- : A, B, A; B, C , . . . . .

k Cn =

n! , k ! = 1 · 2 · ... · k ( k = 0 0! = 1). (n - k)! k! , n · ( n - 1) · ( n - 2) · ... · ( n - k + 1) k , Cn =
k!

() 17 + 27 + 37 + ... + n7 . . n () n ; () k , ; () k , ? k . () , Cn

T1 , T2 , ..., Tn , ... , ) T1 , ) Tn , Tn+1 . n Tn . T1 , Tn Tn+1 -- , Tn -- . . , 11...1 3n .
3n

( ) n, k . , C
3 1/2

=

1 -1 -3 · · 22 2

3! =

1 . 16

() n.
Hint. , .

0 ( ) , Cn = 1 n.

. . , , (. . ) . . , . , , . *. . , . ? *. n . , , .
, , .

. () n . () n . () , 12 + 22 + 32 + ... + n2 = n(n + 1)(2n + 1)/6. () , 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... + n)2 .

() -, 1 . , . k Cn : ? ( .) . n A, B C ? *. . , , . , , , . . , a + 1 , ak + 1k a a , k . *. 12 . , (, , ). , . . ?


A- .


. a, {a} + 1 = 1.
a

(

.)

. [ x ] x , x . x { x } = x - [ x ]. , [2] = 2, [-3, 5] = -4, {-0, 5} = 0, 5, { 2} = 2 - 1. . () , [ x ] = 3, [ y ] = 4. [ x + y ]? () , { x } = 0, 3, { y } = 0, 4. { x + y }?

. xOy , : () [ x + y ] [ x ] + [ y ]; () { x } + { y } = 1; () [ x ] + [ y ] + [ x + y ] [2 x ] + [2 y ]. . :
3 2 2 3 5 + 2· 3 3 2 3 + 2· 2 5 3 5 + 3· 3 3 2 3 + 3· 2 5 3 5 + · · · + 22 3 3 + · · · + 34 5

· 35

23 23 · ? 35



. 0, 1, 2, ..., 6, , . , ? . : () y = [ x ], y = { x }; () y = [ x ] + [- x ], y = { x } + {- x }; () y = 2 x - [ x ], y = [ x ] { x }.
2



. 2002!? . () ,
n + p n p2

= 7 · {/7},

p , n!,

+...

*. ( ) x , () [ x 2 ] - [ x ]2 = 2002; () { x 2 } - { x }2 = 1 .
2002

, () (2n)!! = 2 · 4 · ... · (2n); () (2n + 1)!! = 1 · 3 · ... · (2n + 1)? ( ). n , p1 , ..., pr -- , n. n: 1, 2, 3, ..., n. , p1 , p2 , pr . () ? n? () , n- [n/ pi ] +
1ir 1 i< j r

() () () ( )

.

x , y , z x , y , z x , y , z x , y , z
x y z

() x > 0 n , () x , y , z ,
2

[x ] x n =n.

? [[ x + y ] + z ] = [ x + [ y + z ]]. {{ x + y } + z } = { x + { y + z }}. [[ x · y ] · z ] = [ x · [ y · z ]]. {{ x · y } · z } = { x · { y · z }}.

[n/( pi · p j )] -

1 i < j
[n/( pi · p j · pk )] + ...

... + (-1)r [n/( p1 · p2 · ... · pr )]. . {(2 + 3)2002 }. *. , x1 = 1/2 x
k +1

x = yz . . : () [ x ] x - 1 ; () { x } + {57 x } = 1; () x 4 - 2 x 2 + 5[ x ] - 6 = 0.



1 1 + 1 +...+ . x1 + 1 x2 + 1 x 100 + 1

2 = xk + xk k > 1. -

. [ 1] + [ 2] + [ 3] + ... + [ 10 000].

*. a b, 1, [am ] = [bn ] m, n ?


- .
( .) (Ap- ) . , . . a, b , b = 0. , a . . b ( a . b), c , a = b · c. . . a . b b | a («b a»). . () ? () , a < b, a b? . () a b a . b . . b . a? . . : . . () a . c, (k · a) . c; . . . . . . . . () a . c b . c, (a + b) . c; . . . . c b . c, (a - b) . c; () a . . . . . . . c b . d , (a · b) . (c · d ); ( ) a . . . . . . () a . b b . c, a . c. . . . . , n . . () n(n + 1)(n + 2) . 6; () ((n + 1)3 - (n - 1)3 + 4) . 6. . . (, ) . . a, b , b = 0. a b q r , (q , r -- ). . () -5 -10? -10 -5? () -20 022 002 ? () 12 345 678 987 654 321 ? () 123 456 789 ? () an - 1 am - 1 (a , a 2)? . , , .
Hint. a - b · q.

. , a b . a + b a · b ? *. , , , . ? ? . 1 . () 103993 = 6 +
16551 3+ 1+ 1+ 7+ 1 1 1 1 2+ 1 x

() : 1 +
1+

1 1 1+ 1

(n ).

.. . . :
a0 +
a1 + a2 + 1 1 1 1 .. .+ a
n

,

n 0, a0 , ai : [a0 ; a1 , ..., an ].

i

1.

a = b·q+r



0

r < |b|

. , . ? *. () : [a0 ; a1 , ..., an ] [a0 ; a1 , ..., an+1 ]? () dk = [a0 ; a1 , ..., ak ] -- [a0 ; a1 , ..., an ]. , k 0 d () 87 ;
32
k

d

k +2

d

k +3

d

k +1

.

. : () 98 765 ; () 104 348 .
43 210 33 215

*. () [1; 1, 1, 1, 1, ...]? () 57 .


A- .
( .) . () ( x 5 - 6 x 4 + 3 x 2 + 1)2002 . x . () P ( x ) ; Q ( x ) . , P ( x ) + Q ( x ) P ( x )Q ( x )?
, , . , 0, 1, -1.

... + an x n , P () = a0 + P ( x ),

. P ( x ) -- , P ( x ) = a0 + a1 x + a2 x 2 + ... -- . P ( x ) a1 + a2 2 + ... + an n . P () = 0. . . , P . Q ( P , Q -- ), . Q P . ?

*. P ( x ) , () P (0) = 19, P (1) = 89, P (2) = 1989; () P (1) = 19, P (19) = 89? . () deg P ( «degree») P . () , deg( PQ ) = deg P + deg Q . deg( P + Q ) deg( P - Q ) deg P deg Q ? () , x n () deg( PQ ) = deg P + deg Q . x 2 + x ? *. () P ( x ) P (1), P (2), P (3), ... . , ( P (1) - - P (2), P (2) - P (3), P (3) - P (4), ...). . . , - . () , .

. () , , , ( ) . () , ( , ) . . () x 6 - 31 x - 1 x - 2; () x 5 - 6 x 3 + 2 x 2 - 4 x 2 - x + 1. ( ). () , P ( x ) x - a P (a). . . () a -- P ( x ). , P ( x ) . ( x - a). ? . () P (1) = P (2) = 0. , P ( x ) . ( x - 1)( x - 2). . . () , n- n . () , x = n, x = 2? () ), . . () F ( x ) x - 2 x - 3 . F ( x ) x 2 - 5 x + 6. () x 105 + x + 1 x 2 - 1. . ( [ x ]) () x 5 - 4 x 2 + 24 2 x - 4; () 7 x 2002 + 13 x - 2 x 2 - 1; () x 2002 + x - 2 x 2 + 1; () x 103 - 1057 x 2 + 125 3 x - 1? . , a, b c : 1 1 1 + + = 0; ()
(a - b)(a - c) (b - c)(b - a) (c - a)(c - b) bc ac ab () + + = 1; (a - b)(a - c) (b - c)(b - a) (c - a)(c - b) Hint.

()

b c a + + = 0; (a - b)(a - c) (b - c)(b - a) (c - a)(c - b)

( x - b)( x - c ) ( x - a )( x - c ) ( x - a )( x - b) + + 1. ( a - b)( a - c ) ( b - a )( b - c ) ( c - a )( c - b)

P ( x ) [ x ], P ( x ) = 0, P () = 0 ( ). . . k , P ( x ) . ( x - )k P ( x ) .( x - )k+1 . , . . .


() - a, b, c, d . . (1 + x )
k =0 k C x k .

b2 c2 a2 + + = 1; (a - b)(a - c) (b - c)(b - a) (c - a)(c - b) 3 3 3 b c a + + = a + b + c. () (a - b)(a - c) (b - c)(b - a) (c - a)(c - b)

()

- .
( .) . -- , -- . , -- . , (, ) , (). . , . . , , , . . l A, l , , A l.
, , «» () , . .

. () ( ?) . () , (1 + x ) · (1 + x ) = (1 + x )2 .
. x k .

() 1 + x . () , (1 + x ) · (1 + x ) = (1 + x )+ .

. () ; () . . () -- , , , , . , (, ) -- . () , = â = â ? ( ). , () ; () . . (, ). , () ; () l ; () , 9. . - (, ) -- . l m (l , m ) ( l m), . ( , .) . (, ). () , l m m n, l n l = n.


() , . () , A ( ?) . ( .)


. . ( ?). (, ) k . () , k - 1 . () , k 2 . () ? . () ; () ; () ? . 16 . . , ? ( ). n , n . n2 ( n ) n â n, . () n = 5 ( , , , , , , , ). () , n, . () , n = 14 ( 14 -- ). () , n = 6 ( ) . ( ). .
--, , k 4 1 2 k , k .

. , . . . . . , () ; () ( ); () ; () ( ), . () , ?
, , , , «» «», «» «» . , , : . .

( ). () , . ? () ( ), , . . . ( , , .) () ? () , . ) n + 1 . ) n + 1 . ) n + 1 . ) n + 1 . ) n2 + n + 1 . ) n2 + n + 1 . . , () 3; () 5; () 6 . () , 57 ?


- . - .
( .) . , : (a) ; () . . , ((8 + 4) : (2 - 5)) â (3 : (25 - 22)), , . (()())(()). . n- cn n . cat(s). . () t · cat2 (t ) - cat(t ) + 1 = 0, () cat(t ) =
1 - 1 - 4t . 2t

. (n + 2)-, 0 n + 1. . n . , (n + 2)- 1 cn .

. , 2n cn . . , cat(t ) : cat(t ) =
1 1- t 1- t 1- t

.

. 1 - ..

. , , (1, 0). n n- mn . : m0 = 1, m1 = 1, m2 = 2, m3 = 4.

*. , cat(t ) (. . P /Q , P , Q [t ], Q = 0). . (a) , cn =
(2n)! n = 1 C2n . n!(n + 1)! n+1
1 1 1 2 2 5 5 14 9 14 3 4 5 6 1 1 1 1 1 1

. () 5 . () . . . () 0 N , i - -- ai , -- bi , :
1 . a1 b1 t 2 1- a b t2 1- 2 2 . 1 - ..
. . . . « » (.: , ).

() ( «» ) . . ( . ). () , . () n- ?

.


() ci , :
1 1 - c1 t - a1 b1 t 2 a b t2 1 - c2 t - 2 2 . 1 - ..

*. .
1 0 B
n

.
0 2 2

1 1 1 1 0 E
n

. Up-down a1 , a2 ..., a N 1 N , , , a N -1 > a N . , (4, 7, 6, 9, 1, 3, 2, 8, 5) up-down N = 9. ( !) . * (, ). --. , ( , ) , (n - k )- (n - 1)- -- up-down n, k . *. , :
1 1- t2 22 t 2 1- 32 1- 1-

02455 16 16 14 10 5 0 0 16 32 46 56 61 61 4=2+2 5=4+1 10 = 5 + 5 14 = 4 + 10

.

.
t
2

.. .

y

1

t:: ttt 24 x;; FFF## x x4 x 4 180 ;; 6 F F xx F F## xxx;; xx 3 3 3 28 ;; 2 F xx FFF x x;; FFF## x xx 2 2 ## x 2 2 61 x;; 1 FFF x;; 5 FFF x;; F## x x F## xxx xx x1 x1 1 1 1
1 5

;; xx xx FFF ## 3 ;; 66 xxx

;; xx xx ;; 2F xx FF ## xx 2 2 ;; 1385F FFF ;; xx F ## xxx 1 F## xx 1 1 x 1385 61

.


- . - .
( .)

a0 , a1 , ..., a

A( x ) = a0 + a1 x + a2 x 2 + ... , A( x ) = .

k -1

. , P(x ) , Q(x )

P k - 1. () Q ( x )?

. 2 â 16. 1 â 2? (. K- ) F ( x ), F ( x ) [[ x ]], fn -- n- . . () , (1 - x - x 2 ) F ( x ) = x . x - fn x n+2 - fn () , f0 + f1 x + f2 x 2 + ... + fn x n = 1- x - x . : () f0 + f1 + ... + fn = fn+2 - 1; () f0 + f2 + ... + f2n = f2n+1 - 1, f1 + f3 + ... + f2n-1 = f2n ; () f1 + 2 f2 + ... + nfn = nfn+2 - fn () f12 + f22 + ... + fn2 = fn · fn+1 ; () ( ) fn () fk+n = fk fn+1 + fk-1 fn .
+3 +1 2

.

x

n +1

.

+ 2;

+1 fn-1

2 - fn = (-1)n ;

. () 57? () ( fn ) 57. , . () 57? ( ). ) n- .
2 *. fn ?

( ). () , a2 f2 + + a3 f3 ..., fn -- , ai , ai . () . *. () () . *. . 1 144, , , () «» «». «» 1 , «» -- 2 . () , ? () , 144 ?

() a

. : () an+2 = 4an+1 - 4an , a0 = a1 = 1; () an+3 = 3an+2 - 3an+1 - an , a0 = 1, a1 = a2 = 0;
n +3

= 3a
2

n +2

- 1 an , a0 = 0, a1 = a2 = 1.
2

. () an k c1 , ..., ck : a
n+ k

= c1 a

n + k -1

+ c2 a

n + k -2

+ ... + ck an ,


- . - . Le Bagatelle
( .)
-- l bagatelle? -- . -- Le bagatelle, -- , -- , , , . .

. () «» , ? () m n . , ? () , ? *. . . , , ? ( ). () 2n «». , , -- . ( ) -- . . k - n ? () 4n . . , -- , -- , -- . , , . n k - ( ) S-N l - W-E ? . m n . () ? () (m + n)- , ? *. ( p , q )- (heffalumps) m â n , ? . ( ) .
. p q . , -- (1, 2)-.

. -- fylkestrafikksikkerhetssekretariatsfunksjonene. () ? () . . () a3 e7 f 3 h i 3 j k 7 l n 3 o r 4 s 6 t 4 u y (a + e + f + h + i + j + k + l + n + o + r + s + t + u + y )46 . () : ( x1 + x2 + ... + x k )n =
l1 +l2 +...+lk =n l l l

(...) x11 x22 ... xkk .

( ). () ? () , (- )? . () . . () x1 + x2 + ... + xm = n ? () . ( - .)
.


AWACS, B- , F- C F- A. () ? , () ; () ? . . () , N , n, N n . () ( . ) . () , N m m N + 2 . () , N , n, N , n .



( ) . . . . 2n. n = 3. . . . . a1 x1 + a2 x2 + ... + ak xk = n (ai ) + . . . . , . k . Cn . k Cn n ( ). k C n . . x1 + x2 + ... + xm = n + ( ). . . x1 + x2 + ... + xm = n + ( ). . . . . . . . . , . . . . . . , . . . . . . . . . .

m ( m + 1)

* ( ). (1 - t n ). ( , , .)

:

n =1

.



- . --
( .) A B ( A, B). . ) ( A, B) 0, , A = B; ) ( A , B ) = ( B , A ); ) ( A, B) + ( B, C ) ( A, C ), , B AC (« »). . ( ?) f : 2 2 , , . .
A, B
2

. , () -- ; () , . . () . , . , . () n- , A -- , . , n A. . ( ) l -- , l , X , , X , l -- XX . . , ¨ -- . , () ; () ? . , () -- ; () , , ; () . . l A B. X l , AX BX l () ; () . . () , . () ?
.

( A , B ) = ( f ( A ), f ( B )).

. () , f -- , f -1 ( ?) -- . () g -- , f g -- . () : , -- , . () , . . O -- , O , X X , O XX . . . . , . ?
, .


. () «» , ? () . ; , , . , , . . TA B A B -- , X X , AX BX . ( , B X A BX .) . M N , A B, A M N B A B ? ( , .) . , () -- ; () A B BC AC ; () . . 1 , 0,001. , () 0,34; () 0,287. () 0,22.

- . -- .
( .)

. R O (0 < O < 180) , O O , X X , OX = OX , XOX = OX OX . . () , , . () A B. C , AC A BC , , . . () ( ) . () , -- . () , , -- . () « ... ...» ? . A BC . A B BC A B M N BC PQ . , MQ AC . ( ). A BC , 120 , A BC , A B C A BC . , : () AA , B B CC T ; () T AX + BX + CX ; () P Q -- A B A C , A PQ ; () . . , , 0 < , , 180 + + + = 180 , A, B C . , 2 2 2 RC R B R A , A BC , .


. () A B BC A BC P Q . M -- AC . PQ M . () . , , , . . , . . () , A -- A l , AA l . () A BC A1 B1 C1 -- , A BC , A1 B1 C1 -- . A2 , B2 C2 -- AA1 , B B1 CC1 . , A2 , B2 C2 . : TA B - A B, Sl l R O O . . , TC D TA B = TA E , E = TC D ( B). ( ). , .
TA TC S R
D B E

() 57-, 57 . () A , B , C . ( ). , () ; () ; () ; () . () , 57 , . () 3 â 3. ( ) ?

S

l1

R

O1

TA

l2

O2

. , () , ; () . ( ). , . *. () 2003- .


- .
( .)

. a, b , a b . a b , a b. : (a, b). . () (n, n + 1); () (n, n + 6); () (2n + 3, 7n + 6); n m () (2n - 1, n2 ); () ( ) (22 + 1, 22 + 1).

. a b (a > b) b. , . () 6188 â 4709? () , . . () ? . , : () (k a, k b) = k (a, b); () (4a + 2b, 6a + 4b) = 2(a, b); . . () (k a + lb, ma + nb) . (a, b); () (13a + 10b, 9a + 7b) = (a, b); () (Sl2 ( )) k n - ml = 1, (k a + lb, ma + nb) = (a, b) ( k , l , m , n ). () (k a + lb, ma + nb) (a, b) : k n - ml = d ? (k , l , m, n .)

. () . (a, b), b = 0, (b, r ), r -- a b. r = 0, , (b, r ). . , () (a, b) , b = 0, - ; () d = (a, b), (d , 0) -- , . . : () (81 719, 52 003), (6188, 4709), (-315, 159);
1001 57

() (11...1, 11...1). *. () , . , . () , , a b, , . = a/b ( ). () , log ( 5(max(a, b) + 1/2)) - 1.
( E ), III . ., -- . «» , ( ). « », « » , , . « » (, , ), -- .

*. n, 1. , . , -- 1, -- . , , . , ?

(). () . ? ( «», ? «», ?) () ( ) , a, b (a, b) = = d , k , l , k a + lb = d .

*. p q ( p , q )- ?

*. : k > 1 , , . - .


. , , . ?
(, , ...), (?). , -- , .

- .
( .)
, , , .

a a

. M -- , a, b M , b . b d , . . . . . : (a . d )&(b . d ) (d . d ). . a b (a, b). . () ,


(a, b) = {±



(a, b)}.

() ? () 2 , 3 . () , [ x ] ( ) , . ? . [ x ]: () x 2 - 4 x + 3 2 x 2 + 4 x - 6; () 2 x 4 + x 3 - 1 x 2 + 1 x 2 - x + 1 ; () x n - 1 x m - 1, n, m .
3 4 2 5

. «» -- xOy , S2 -- O R, N = (0; 0; R) -- « ». N . ( ) N : S2 \ { N } xOy , xOy , -- , (. . ).
N m 0
2


S

ò N (m)

.

. - .

. () N R = 1 : S ( ), (1/3; -2/3; 2/3), (0; cos ; sin )? () N 1 : (2; 0), (1; 2), (5; 7)? () 1 , 1 1 , , . . () . () ? ?


() «--» ( 1). () , N ( x ; y ; z ) = () -1 (u; v ). N . . ( ?) . O R InvO, R = S -1 , S -- N . . () , O . , . () . () -1 Inv N : N S2 - -- -
N

, () , () . . () . . () A B S1 S2 , S, S3 , S. , S3 S1 S2 . . () , . () ( .) , . () l M , A B, l , a. . () , , -- . () R1 R2 S1 , ..., Sn -- , R1 R2 ( ) . , T1 , R1 -- R2 , , . ( ). , : () , , () , n , () , () 1 , n () , () , () A B, () A B C D . ( --). , , , . ( , -- , -- .)

Ry Rx ; . R-z R-z

?

xOy - -- xOy ? - . 1 () (0; 0), (1; 0), (0; 4), (3; 4), (-1; 6), (0, 01; -0, 01), (2 sin ; 2 cos ); () : x + y = 2, y = 3 x + 1; () (0; 2), (4; 0) (5; 1); () ; () (-3; 4) 5 , y = 4; () « » (3; 1) 3; () x = const y = const? . O . , () O ; () O ? () ?

Inv

S2

N


. P1 A B = b A P2 AC = c. S0 BC c - b P1 P2 . S1 , S2 , ... , Sn P1 , P2 Sn-1 . Sn AC . . «» , . . , () ; () «» ; () . . , . , .

- .
( .) . p = ±1 , p = ab (a, b ) , a, b ±1. ( ). () , . , () 4n + 3; () ; () 4n + 1.
, , a0 d .

. p , () p + 57; () p + 4 p + 14; () p 4 + 4 -- ().
p p + 2 -- , -. -, -- .

. () , . () , , n, . () n, O ( n) . . a, n , a, n 2. () , an - 1 -- (a, n 2), a = 2 n -- . . . . () , an + 1 -- (a, n 2), a . 2 n = 2k . k 2 2 + 1 . () ?
, .

. a b , , ±1. . a b -- . , () x , y , a x + by = 1;


. . () (a · c) . b c . b; . . . . . . . () p -- , (c · d ) . p (c . p ) (d . p ). . (). , () n n = ± p1 1p2 2 ... k ... pk , pi -- , i , n; () ( ). . . () ? () n = ± p1 1 p2 2 ... pk k . n? . () . -- ? () , (a, b) · (a, b) = a · b. . , () 2 -- (. . 2 = m , n -- n m -- ) ; () k a (a, k ) -- , a = bk , b . () x n = y m ( x , y , n, m ), n m -- . , t , x = t m , y = t n . () x y = y x . . () , () p n! [n/ p ] + [n/ p 2 ] + [n/ p 3 ] + ... ... + [n/ p k ] + ... . () 2003!. . () , n! . 2n . . () - ,
n k

() ( m x n x = (mn) x ). (. ). : ( x ) = 1 + 1/2 x + 1/3 x + 1/4 x + ... - . *. () 2 ( x ); () ( x ) · ( x - 1); () L( x ) · ( x ), L( x ) = 1 - 1x + 1x - 1x ± ...;
3 5 7

( ) -1 ( x ) .

*. , () 2 ( x ) = (n)/n x , (n) -- n; () ( x ) · ( x - 1) = (n)/n x , (n) -- n; () 4 L( x )( x ) = M (n)/n x , M (n) -- n. () n .
n, ( n), . . (n) = 2n, . , 6, 28, 496. , .

( x ) =

*. , () ( )
p

1/(1 - p

-x

)=
p

(1 + 1/ p x + 1/ p

2x

+ 1/ p

3x

+ ... ),

p ; () L( x ) =
p =4 k +1

1/(1 - 1/ p x )

1/(1 + 1/ p x ).
p =4 k +3

=

-- . () ,

n! (n - k)! k!

. , .

p k

. . p , p -- . , p -- .
(Riemann), -- , -- , , , . , , - , . , - Re z = 1/2, .

? . k1 + k2 /2 x + k3 /3 x + k4 /4 x + ... () . , 1.


- .
( .)

. (a, b) (a, b ). (a, b) + (c, d ) = (a + c, b + d ) (a, b) · (c, d ) = (ac - bd , ad + bc). [i ]. . () , (0, 1)2 = (-1, 0). (0, 1) i , (a, b) -- a + bi . () , , . . a + bi i 2 = -1. ¯ . z = a + bi . ¯ = a - bi z z , N (z ) = a2 + b2 -- z . . : ¯ ¯ () z + w = ¯ + w ; z · w = ¯ · w ; z z . ¯. z. () z . w , ¯ . w ; . ¯ () N (zw ) = N (z ) N (w ); N (z ) = z¯; z ¯ ¯ () z + ¯ z¯ -- . z z () 1 + i [i ]. () [i ]. . : () i n (-i )n ; () N ((1 + i )n ) N ((1 - i )n ).

. (z , w [i ]) : ¯ () z = ¯; () z = N (z ); () z · (2 + 3i ) = 3 - 2i ; z () z + w = zw = 2; () N (z ) = 49, N (z ) = 57, N (z ) = 55. . () ( ), , x 2 + y 2 . () ; () ? , ( ) x 2 + 2 y 2 , () x 2 + 5 y 2 ? . , () v z = vk (k [i ]) ; () N (v ). () ( ?). , z . q , (z - q ) . v . . . , [i ] , . . u, v [i ], v = 0, w r , u = vw + r , N (r ) < N (v ). [i ]? . [i ].
[i] -- a + bi -- n [ -n] -- a + b -n. -

. z , z = ab (a, b [i ]) , a, b . . [i ]: () i ; () ; () 5i ; () 3 + 2i ; () ; () 7 + i ; ( ) 7 + 8 i ?

. () z ¯; () z i z ? z

. () () [ -n]. () . () n . () [i ], [ -2], [ -5] ( ). () ( ) () [ -2], [ -3], [ -5]?

. , : () N (z ) -- () , z -- () . ? ¯ () z [i ] -- ¯ -- . ? z
a, b , ´ , . , u K K , v K , u · v = v · u = 1.


- .
( .)

() , 17i - 41, , 27 - 35i .
m ( .), «» . ´ . .

. () , , ., ., . . ( ) « », . -- . ? () ? . a b m, . (a - b) . m. : a b (mod m). .

. () a, a -2 (mod 7); a 1000 (mod 24); a 256 (mod (-4)); () u , u -i (mod 1 + 2i ); u 2 - 3i (mod i ). () , () P ( x ) , P 2004 (mod x 2 ); P x + 1 (mod ( x + 1)).

. , () a b (mod m) a b m; () m m; () w N (w ); () , , , ;

. : () -- ; () a b (mod m) c d (mod m), ac bd (mod m); () P [ x ] a b (mod m), P (a) P (b) (mod m); () p -- , (a + b) p a p + b p (mod p ) ( p , a, b ); p ? () ( ) a p a (mod p ) p .

. m -- , m > 1. m {0, 1, 2, ..., m - 1} m, ¯ -- n m. a, b m «» n ¯ «»: a + b = a + b, a · b = a · b. . () , ¯ = ¯ a b (mod m). ab ¯¯ () 4 ; () 7 ; () 8 ; () 9 . . , , () ; () . . , . () (n5 - 5n3 + 4n) . 120 (n ); . . 2222 () (5555 + 22225555 ) . 7; . 2 n +1 n +2 n +2 2 n +1 . () (5 ·2 +3 ·2 ) . 19 (n ); . . () 157 + 257 + ... + 100057 . 1001; . . . () (a2 + b2 ) . 7 (a, b ) (a2 + b2 ) . 49 (a, b ); . . () , 4k + 1 (k ). 14 . () 1414 . () n 13- 21 982 145 917 308 330 487 013 369. n? . () ; () ; () ; () . . () , x 12 - 57 x 7 + 91 = 0 . () P ( x ) [ x ], P (0) P (1) . , P ( x ) = 0 .
n . . . ki (|ki | n/2), an an-1 ...a0 . n (an kn + an-1 kn-1 + ... + a0 k0 ) . n. , . 3 ki 1. , .


() , 7 x 2 + 2 = y 3 ; () , x 5 + y 5 + z 5 + t 5 = 5 . . a K ( K -- ). a , a = 0 b K , b = 0, a · b = 0. , . . , () [ x ], [ x ]; () [[ x ]]; ) [i ]. () m m ? () 6 [ x ]?
a x b (mod m), . . m , , , a x - my = b.

- .
( .) . () , . () n n- ? . . L. : (a, b) + (c, d ) = (a + c, b + d ). . A : L L, A( P + Q ) = A( P ) + A(Q ) P , Q L. . . def () A(0) = 0 ( 0 = (0, 0) -- ), A(- P ) = - A( P ). () A(n · P ) = n · A( P ), n . () A A(1, 0) A(0, 1). () A ( x , y ) = ( a x + b y , c x + d y ), () , ad - bc = 0. () , ad - bc = ±1. . () POQ R L. , |ad - bc|, P = (a, b), Q = (c, d ). (O -- .) () . , 1 c . () , . , . ( ). , S S = n + m - 1, n --
2
def

. d = (a, m). , () a x b (mod m) , . b . d ; . . () b . d , d m ; . x0 -- , -- m = m/d . () x0 . () 29 x - 13 y = 2. () a1 x1 + a2 x2 + ... + an xn b (mod m). x0 + m , x0 + 2m , ..., x0 + (d - 1)m ,

a, b, c, d .


, m -- . . L . , ad - bc = 1, Sl2 ( ). . , : () A, B Sl2 ( ) A B Sl2 ( ); () A Sl2 ( ) A-1 Sl2 ( ).
, Sl2 ( ) .

, -- . 1 . , . . m -- . [ m] (a, b) ( ), : (a, b) · (c, d ) = (ac + mbd , ad + bc). (a, b) a + b m. [ m] N (a + b m) = a2 - mb2 . . z , w [ m]. , () N (zw ) = N (z ) N (w ); () N (z ) = 0 z = 0; N (z ) = ±1 z ; () a1 a2 (mod N ), b1 b2 (mod N ), . . (a2 + b2 m), N = N (a2 + b2 m). (a1 + b1 m) . . () Mc , x 2 - my 2 = c. () , Mc -- . () x = ± m y , Mc M-c . . ( ). , : () c | x 2 - my 2 | < c ;
Hint. Mc M .
-c

() : L . () ? . () m n . ? () L : (a1 , b1 ) (a2 , b2 ). ? , (0, 0). () , 1 Sl2 ( ) . () , , . , . , . . Sl2 ( )- () ; () . . S. , () S < 1, , ; () S , , S , -- S ; () ( ) , S > 4, ( ). . 1 1 . -



()

() c c; () N (z ) = 1 [ m ] , ±1; () z0 = a0 + b0 m [ m ], n 1 ± z0 , n . a0 , b0 > 0 z0 x 2 - my 2 = 1. : () x 2 - 5 y 2 = 1, () x 2 - 41 y 2 = 1.

m d 2 d > 1.


- .
( .) . a p ( p -- ). p - 1 , p ( ). x a · x . a. () p = 13, a = 5; p = 7, a = 3. , : () ; () ; () ; () a p , a . () a Zm m? . () m m ? () -- m m m. () m . m = 30, a = 7. () , 1 + 1/2 + 1/3 + ... + 1/( p - 1) p ( p -- ). ( ). a -- . , . () p -- , a . p n : an 1 (mod p ); . . n a p ord p (a). . () an 1 (mod p ) n . ord p (a) (n ). . () ( ) (a, m) = 1 a(m) 1 (mod m), (m) -- 1 |m|, m. ( , , , (m) .) ( --). p > 2 -- . , () 111...11, p ( p = 5);
, , .

. , 0001? * ( ). , 561 : (a, 561) = 1 a
560

() 1/ p p - 1 ( p = 5); () 2 p - 1 2k p + 1.

1 (mod 561).

, , . *. p > 2 -- . () x p-1 - 1 p . . () , ( p - 1) . d , x d 1 (mod p ) . d p . () , p a, ord p (a) = p - 1. ( , .) . ( ) n- , k = 1 k , 1 k < n. . () , n- . () , , , . () , n- (n). *. k - n- . ( ). () 4 , 5 , 6 ? p -- . () , x 2 = a p ; a ? () p ? () , x 2 + a x + b = 0 p , a2 - 4b -- p . . . p > 2 -- . : () () ( p - 1)! -1 (mod p ) p -- ;

an 1 (mod p ),


, -: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)...

() () ( p - 2)! 1 (mod p ) p -- ; () ( ) p p + 2 - , 4(( p - 1)! + 1) + p 0 (mod p 2 + 2 p ). ). p > 2 -- . -- ; -- ; -- ; p -1/2 1, -1; p ) a




( , p () () () ( ) a = 0 ( a

.
( .)

. a b, . n . . () , . () , . . : () 3m + 7 = 2n ; () 3 · 2m + 1 = n2 ; () 1 + 1 + 1 = 1.
a b c a+b = 3. 13 a 2 - a b + b2

() x p-1/2 = 1 x p-1/2 = -1 2 ; () a = 0 , a p-1/2 = 1. () p 2 p ?

p-1

. 3n2 + n + 1 n? . , . , , , , . ? ( ). m1 , ..., mn , m = m1 · ... · mn . () , x b (mod m) x b (mod m1 ), ................ x b (mod mn ). () , x b1 (mod m1 ), ................. x bn (mod mn ).

« ! -- . -- : ? : !»

, ,

m

b1 , ..., bn .


, m = m1 m2 ... mn . () b1 = 1, b2 = b3 = ... = bn = 0.

( ). . ( 1 37) , , . , , , , .
, 9 3â ; , 2 â 2; , .

1 , ..., (n) -- 1 n. () n ( x ) 12 n. () , n ( x ) . ? () x n - 1 = d ( x ). () n ( x ). () , n ( x ) . *. () , , . S(n ) < 1 , () , n > 108
n 20
d|n

. () , a a mi . , = 1 â ... â n . m m m () 004 ? 771 ? 2 1

m

,

S(n) -- , n.

() ord2004 (5), ord1771 (16). . () ( p k ), p . () , (m) = (m1 ) · ... · (mn ) mi . () (n). *. : an : 1 3 4 6 8 9 11 12 14 16 ... bn : 2 5 7 10 13 15 18 20 23 26 ... () , . n () : an = 2 bn . () 1 2 4 5 7 8 9 11 12 14 ... 3 6 10 13 17 20 23 27 30 34 ...
Hint. ,
1 5+1 2 1 5+1 +1 2

5+1

+

= 1.



. n ( x ) ( x - 1 ) · ... · ( x -
(n)

),


A- .
(«» . / ) . . . . [ x ]. . . . , . . (). [ x ]. . . . . [ x ] [ x ]. . . . . [i ]. . . [i ]. . a x + by = d . . . . . m , . . . m . . . ( ). . ( m ). . . ( ). . p . , , . ( .)
-- , , , () .

. () , . . ( ) +: : ) «» ( «+») 0, x x + 0 = 0 + x = x. ) x
y , , â

,

x + y = y + x = 0.

y x «- x ». ) «+» -- , . . x , y , z x + ( y + z) = ( x + y ) + z. x + y = y + x. ) «+» -- , . . x , y . -- . , ; . , (, -, ). . ( )
·:

, :

â



,


) () « · » -- , . . x , y , z x · ( y · z) = ( x · y ) · z;

) ( -- ) , . . x , y , z ( x + y ) · z = x · z + y · z;

. . X , C , x X x C . C X . ( ). : . c X , : ) c -- X ; ) C X , c C.

) ( ) \ {0} «» ( « · ») 1, ) ( -- ) x \ {0} y \ {0}, x · y = y · x = 1.
x \ {0}

x · 1 = 1 · x = x;

y x x ) ( -- ) « · » -- , . . x , y x · y = y · x.

-1

.

: c = sup X , : c = sup x . , c X , > 0 x X c - < x ( c) (!). . () X c . ( )­( ) 1 , . . ; 2 , , , . . , , , .
x X

, , . -- . . -- . . , , , p ( p -- ) , , , [ x ], 57 -- (?). . , . . « », : ) x ( x x ), ) x , y ( x y ) ( y x ) ( x = y ), ) x , y , z ( x y ) ( y z ) ( x z ). , . -- . , . , . . ) x , y ( x y ) ( y x ), . .
x, y, z

(x

y) (x + z

y + z ).

. . , , p (?). (?). (?).
x, y

(0

x ) (0

y ) (0

x · y ).


A- .
( .)
«You may call it `nonsense' if you llke», she said, «but I've heard nonsense, compared with which that would be as sensible as a dictionar y!» Lewis Carroll. Through the Looking Glass . , .

(/ ). () , , . , a + x = b . () . () , ( «») . () , (- x ) = (-1) x , (-1)2 = 1. ( ). () , >, <. ? ( , .) () , x y : x < y , x = y , x > y . () , 1 > 0. () , . ? ?
, , , -- 1, 2, 3, ... .

. , 1. . () , . ( « » .) () . ( ). () (.) : M x x + 1, M = . () , ? () , Ap- . () , - , . () 2 â 2 = 4 -- ? , , ? ( ). () , 57 -- . () , -- . () , n n + 1 (n ) .

, , .

( ). () ( ) , . () ( ) , .
. , .

( ). () : . , -- . . X , x X x +1 X.
; A B (a A a B).
def



( ). () x = sup{t | t > 0 t 2 < 2}.

, x x = 2 ( ).
, : 57 = 56 + 1.
def


() , ( ). ( ). () . () ( .) , : . , . () ( ), .
h > 0 x
+

- .
( .)
. , . , , , . .

! n

: (n - 1)h < x

n h.

. a : . a(1), a(2), ..., a(n), ... a a1 , a2 , ..., an , ..., (an ).
, . M , a : M .

. , an+1 > an . n , C n an < C . . () , , , . . , , , , ; , . () , . . , : () an = 1 + x + x 2 + ... + x n , | x | < 1; () an = (-1)n n n;
1·2

() an = 1 + 1 + ... +
( A µ), -- . . ., -- , .

1 ; 2·3 n ( n + 1) () an = 1 + 12 + 12 + ... + 12 ; 2 3 n


() a n = 1 + 1 + 1 + ... + 1 ;
2! 3! n!

() an = 2 + 2 + ... + 2;
n

( ) a n = 1 + 1
n

n

(: );

() an =



n! . nn 2

. () , - . () , , , . () , . . , () ; () , +, - ; () . . : () an = (-1)n ; () an = sin n ; () an = sin n; () an = cos n + sin n + 1 · cos n · sin n .
3 5 n 7 11

. , , + (an ), C {n | an > C } .

. , : () an = 1 + x + x 2 + ... + x n , | x | > 1; () an = 1 + 1 + 1 + ... + 1 ( );
23 n 1111 () an = 1 + + + + + ...; 3579 () a1 = 1, an+1 = an + 1 ; an

() an = :

n

n!.

. , () an = 1 + 1 n
n

; () an =

n

n, n

3.

. (ni ) -- . (ai ), ai = ani , ~ ~ (an ). , 1, 3, 5, 7, 9, ... 1, 2, 3, 4, ..., 1, 1, 2, 3, ... 3, 1, 5, 7, ... -- . . () , () -- (). () , . () , . *. , . . , A (an ), > 0 {n | |an - A| < } -

. ( x - , x + ) ( > 0) - x U ( x ). . A (an ), > 0 , , U ( A). : an A lim an = A.
n

: A) lim an = A U ( A) N
n def

n > N a n U ( A ); n > N d (an , A) < , d ( x , y ) -- n > N |a n - A| < .

B) lim an = A > 0 N n x y ; C) lim an = A > 0 N
n def

def

.

, , , , (« »), ( ) -- , (« »).


. () ? , , , ? () ( .) , an A an B A = B. () 57 ;
(-1)n ; () sin n ; () cos n; n n n () 1 + 1 + 1 + ... + 1 ; () n + 1 - n ; 23 n 1 ( ); ( ) n q 2 () 1 + q + q 2 + ... + q n ; () n + n + 2 . 2 n - 3n + 4

- .
( .) . () , . . () an . , an . () , an , ? () an 0, (bn ) ; , an bn 0. . lim an = a. , () c
n n

. (an ), an

() 1 +

lim (a

n +1

- an ) = 0.

. , : () N > 0 n > N |an - A| < ( ); () > 0 N n > N |an - A| < ? () « (an ) ».

() lim |an | = |a|; () k

n

lim can = ca; = ak .

n

( ). n- (n ) . , lim an = a an 0,
n

lim ak n n

lim

k

an =

k

a k .
N n > N |a n | > C .

n

NB: .
() (. . ). , {} = S1 ().

C

. an ,

N = 57
-5 -5

57

.

a). () , lim an = ( lim an = +) ( lim an = -) ?
n n n

an = 0?) () lim an = + lim an = - n n

. () , an 1 0. ( -
a
n

± ±/2 [-/2, /2] t arctg t (. . ). , {±} = .


y

y y = arctg x

ò/2 = +

y=x

57

0

x

- ò /2 = -

0

1 k -1 nn

k n -1 nn

1

x

.

.

:

( ). lim an = a, lim bn = b. , n n

() lim (an + bn ) = a + b;
n

() lim an bn = ab.
a

() b = 0 n bn = 0,
n bn

lim

n

= a.
b

. : n () lim an (a = const); () lim 2;
n n
3 () lim n + n ; n 2 - 3n

() lim

n

() lim (2n + 3n + 4n )/(5n + 6n ); ( . ); () lim n sin n;
n n ) lim (157 n

4 -2 () lim 42n 3 4n2 + 9 ; n 6n - n - n - n - 1

5n ; 7 n 2 - 2 n - 93

( ). () , . () , , . () , A. , (. . ). , ? () , . ( ). an bn -- ;
n

1 58

lim an = A,

+ 257 + ... + n57 )/n

(
0

x

57

d x -- .

n

lim bn = B.

() lim n ( = const);
n

() lim

n

() lim

. () 1 . ? () 1 . , 10 % ?
, b = 0 (. . , ) bn .

n

f (n ) , f g -- . g (n )

n

n;

, : () N , n > N an bn , A B; () A < B, N , n > N an < bn . () , a) « » «<»; ) «<» « »? () ( .) , an cn A n > N an bn cn , bn A. a + a + ... + an *. () lim an = A lim 1 2 = A. n n

n

? () .


() () lim

a

n

. : () lim ((n + 1)100 - n100 ); () lim (
n
3

n an-1

= , () lim
n+1

n

n

an = . (

- .
.)

n

100 -

100

n);

() lim

() lim 1,00000001n /n () lim n


n

n 2 + 57 ; n + 14
2004

() lim (sin 1 + sin 2 + ... + sin n)/ n;
n 2004

n

() lim {( 3 + 1) } ({} -- ); () lim ( lim cosm (2 x · n!)).
n n m

n

n!

;

n () lim n 1 ; () lim n n ; () lim ak ; lim a ; n

; () lim 0,99999999n · n
n n

2004

;
n

. (an ) ,
> 0 N n > N m > N |a n - a m | < .

n

n

n!

n!

n

n

n!

. , . . . A, B , A B (. . a A, b B a b), c (. . a A, b B a c b). . . . ( .) [a1 , b1 ] [a2 , b2 ] ... [an , bn ] ... , . . ( --.) . . . . ( .) . . , ( ): + + ( -- ). . , () ( ); () , ; () , . *. , - , , , .

*. () , , :
n

lim

2i , c , i 2i , c , i

n > 1. n

-: i 2i 7, 8. () 2i , 8; . ?
. 2i (. ()) (. ()) , log10 2 = 0,3010... 3/10.


. , : () an+1 = p + an , p 0, a1 - p ; () a
n +1

=

() an+1 = sin an , a1 -- ; () a1 = 1, a2 = 5 , an = 3 an-1 - 1 a
2 2 2 a a

p an + ,p 2 2 an

0, a

1

0;

() , , . () , , p + q d , p , q , d . () , . ? ( e). , :
n

n -2

n

3.

m + 1 + ... + nn m, a1 ...an . 10 10 () , bn = m, a1 ...an -- . m, a1 ...an ... m, a1 ...an ... . () . ? () , , (. . N , k : n > N an+k = = a n ). () , . ? * ( ). (), n1 +
n2 + n3 + 1 1

( ). m , m 0, (ai ), ai {0, 1, ..., 9}. -

() 1 + 1 n e; () 1 + 2
n
n

;

() e = lim 1 + 1 + 1 + ... + 1 ; 1! 2! n! n 0! () 2 < e < 3; () e - 1 + 1 + 1 + ... + 1
2! n!

e2 , 1 - 1

n

n

?

=

(0, 1). () e . () , e . () e .

n , n -- n! n

()
1

.. .
+ n + 1 nk

k -1

n1 , n2 , ..., nk , () , [n1 ; n2 , ..., nk ]. () q1 , ..., qk -- , (a, b). , [q1 ; q2 , ..., qk ] = a . () m (ai ). , bn = [m; a1 , ..., an ] . (: b2n < b2n+2 < b2n+3 < b2n+1 .) [m; a1 , ..., an , ...]. () ?
b

. -- ( ). : || -- .

( ). P -- , Q -- . () , sup P inf Q . () , sup P = inf Q . () . () , . (, , ). () , 3 < < 4. () Pn -- , R ( ), dn -- . , , dn 0, Pn 2 R.

, e , , . . .


. () ( ?) 1 2 . , || = |1 | + |2 |. () , . () , x (0, /2) sin x < x . *. 2n · n .

- .
( .) . lim xn = a, lim yn = b.
n n n

2-

2+

2+...+
n

2+

3

lim

xn y1 + x

n-1 y2

+ ... + x1 yn

n

.

. 0 < a < b. (an ), (bn ) : a1 = a, b1 = b, a
n +1

( ). S -- , T -- . () , sup S inf T . () , sup S = inf T . () . () Sn -- n-, R. , Sn sup S. () , R R2 . () , x (0, /2) x < tg x .

=

an bn , bn

+1

=

(an ) (bn ) .

an + bn . , 2

. ( xn ) -- . () , l -- L -- . () , ( xn ) , l = L. () , L = lim sup xk , l = lim inf xk . () , lim ( x
n n k >n n +1

[l , L] ( xn ). ( xn ) .
n, m
n

- xn ) = 0,
n +1

n k >n

. , lim ( xn /2 - x

) = 0. , -

. ( xn ), 0 x
n+ m

xn + xm .

, ( xn /n) . . () lim sin 2 n2 + n ; () lim n sin(2en!). . ( xn ) -- tg x = x , .
n n n

lim ( x

n +1

- x n ).

. a0 = 57, a

n +1

= arctg(an ). lim an = ?
n

. fn ( x ), x . , g( x ),


k xn
n

lim

fk ( xn ) = 0. g ( xn )
i

- . .
( .) D ( f ) f ( D ( f ) ), ° U (a) -- a: ° U (a) = { x | | x - a| < , x = a} ( > 0). (). a -- D ( f ), . . , a, D ( f ). , a D ( f ). A f ( x ) x a,
> 0 > 0 : x D( f )
x a

. : a1 , b1 , c1 . ai , bi , c : a
n +1

= (bn + cn )/2,

bn

+1

= (cn + an )/2,

c

n +1

= (an + bn )/2.

() . () m- n- . . xy = 1 A
n

xn =

n n+1



B

n

xn = n + 1 .
n

Mn -- , An , Bn . lim Mn . . , , , . . () 2000 O ? () ?
n

(0 < | x - a | < ) (| f ( x ) - A | < ).

: A = lim f ( x ). . () lim 2 x = 4. () ( x ) x 0, ( ( x ) -- ). () lim x sin 1 = 0 (. )). . () «| x - a| > 0» «a -- D ( f )»? () , : ° lim f ( x ) = A V ( A) U
x a D( f ) x 0 x 2

x

° ( a) : f ( U

D( f )

( a )) V ( A ).

( ). a -- D ( f ). ( lim f ( x ) = A) ( lim f ( x ) = B) A = B.
x a x a

. . . () (, . .) , ( ) .


() , . . (a) , . () lim f ( x ) = 0, g( x ) a ( a D ( f ) D ( g)), lim f ( x ) · g( x ) = 0. . . ( ). a -- D ( f ) D ( g), lim f ( x ) = A, lim g( x ) = B. () lim ( f + g)( x ) = A + B, lim (k · f )( x ) = k · A; () lim ( f · g) = A · B;
x a x a x a x a x a x a x

lim f ( x ),

x +

lim f ( x ),

x -

lim f ( x )

° U (a) > 0, ° (a) D ( f ) f ( x ) > ( f ( x ) < - ) ( x U ). () lim f ( x ) = A, lim g( x ) = B, A < B, ° ° U (a), x U (a) ( D ( f ) D ( g)) f ( x ) < g( x ). () a -- D ( f ) D ( g), lim f ( x ) = A,
x a x a x a x a

lim g( x ) = B,

° U (a) , ° x U ( a ) ( D ( f ) D ( g )) A B. f (x) g ( x ),

. , , . . ( ). . ( ). ,
x 0

() B = 0, f (x a g( x ) f (x ) lim = A. ) B x a g ( x ) , lim( f + g)( x ) = lim f ( x ) + lim g( x )
x a x a x a

.

lim sin x = 1.
x

. () , . () lim ( f ( x ) - A) · g( x ).
x a

( ). a -- D ( f ), lim f ( x ) = A , ( xn ), xn D ( f ) \ {a}, , xn a f ( xn ) A. . . ( e ). , e = lim 1 + 1
x x x a

. . ) . . : () lim sin(2 x ); () lim sign( x ) («signum» -- ); x2 - 4 . 2- x -2 x 2 x ( ). () () lim lim f ( x ) > 0 ( lim f ( x ) < 0),
x a x a x /3 x 0

x

.

«» , , , . ( A ), . sign( x ) = 1 x > 0, sign( x ) = -1 x < 0, sign(0) = 0.

( ). a -- D ( f ) (-; a), f ( x ) a. f ( x ) x a - 0 , f ( x ) a. .


. lim f ( x ) = sup f ( x ).
x a x U
-



1 = 0.

. lim f ( x ) = ±, x a

- . .
( .) . x , x .

. lim

. : () lim n x ; () lim ( 3 x - 1)/( x - 1); () lim x ( = const);
x + x a x 1
x () lim a k . x + x

x

f (x ) , f g -- . g( x )

() lim a x (a = const);
x 0

, , , , .

. f M , M D ( f ). f M ( f , M ) = sup | f ( x1 ) - f ( x2 )|.
x1 , x2 M

D ( f ). > 0

* ( ). a -- f x a , ° ° U ( a ) ( f , ( U ( a ) D ( f ))) < .

. x M : , M ; , M , x ; , M \ M ; , x M x M . M (Int M ), -- ( M ). . , , : () [0; 1]; () {(-2)/(2 + n) : n }; () (0; 1). . . M ( ), (. . M = Int M ). M ( ), (. . M = \ ( ). . a) , , , -- . ) . . , () ; () ; () ;


() . () , , () () .
, X , X () (). X .

() . (). M U , M U . M

(). , . . . , . . - , -- . . M M ( M ). . () [a, b] = [a, b]; () (a, b) = [a, b]; () = . . , () ; () M , M = M . . , () M -- , M ; () M -- , M . . , () M = M M ; () Int M = M \ M ; () Int(Int M ) = Int M ; () \ M = \ Int M ; () Int( \ M ) = \ M .

( ), (. . , - M ) . . , () -- ; () -- ; () -- ; () -- ; () ; () -- . (-- ). -- . . , . . . . . . , . . *. , -- . . M , M , . , () , , ; () , ( ) , . () , () . () () () ? * ( ). , M , () M -- ; () M M .

. M , M , M , Int M , Int M , Int M , Int(Int M ) () ; () ; () . . , . ( ). [0, 1] (1/3, 2/3) ( ), . . , () 1; () ( , ) 0 1 ; () , ;




- . .
( .)
f ( ) a, a f (a).

(). a -- D ( f ), f a , lim f ( x ) = f (a). a -- D ( f ), f a. . , f a, () a; () f (a) = 0, f a. . f g a, f + g, f · g a. g(a) = 0, f / g U D ( f ) D ( g) a.
x a

. f a (a D ( f ) ) , f M ( M D ( f )), M . M C ( M ). . () f ( x ) = 57; f ( x ) = x ; f ( x ) = 1/ x , . () \ {0} . . , : 1, x , () () ( x ) = 0, x \ ; () () (x) = () f ( x ) = | x |. . , - - (. - ), . ( ). f : : () f ; () ; () . . (a) () , () () .
1 , x = m n n 0, x \ ;
m -- , n

> 0 > 0

f (U (a) D ( f )) U ( f (a)).

. D ( f ) D ( g) 2 A- , .

( ). ) ( 1 (a)) . ) , (). ) sin x cos x , tg( x ) ctg( x ) -- , .
= | x - x0 |.) (: | sin x - sin x0 | = 2 cos
n n

x-x x + x0 s in 2 2

0

2 s in

x-x 2

0

2

x-x 2

0

=

()
x a

x , .
x=
n

lim

a -

. a = 0 -- .

() a x .
a > 0: lim a x = a x0 lim a
x x0 x x0 x - x0

= 1 lim at = 1 lim
t0

n

n

a = 1.

. . ( ). a D ( f ), f a. a f . () « a -- f » , . lim f ( x ), a f (?).
x a

M , M , . . M . M .


() . f (a + 0) = lim f ( x )
x a +0 def

() S :=

.
x a -0

x

sup

f ( x ). g( x ) =

1 S - f (x )

f (a - 0) = lim f ( x )

def

f (a), a f . () . -- f , f . () . () ? () , . () , . . f a, g -- f (a), ( g f ( x )) = g( f ( x )) a. . W ( g( f (a)) g( f (a)). g V ( f (a)), f -- U (a), g( f (U ) D ( f )) W . f ( U D ( f )) V .
sin( x 2 -5 x )

. . *. , () f -- , () -- . (--). f C ([a, b]), f (a) < 0, f (b) > 0, [a, b] x , f ( x ) = 0. . , x . , x [a, b]. , f ( x ) 0 ( ) f ( x ) 0 ( ). ( ). f C ([a, b]), f ( a) = A, f (b) = B,

f C A B. . () , . () , [0, 1] (. e. f ( x ) = x ).
, , , , ( )... .

. g( x ) = f ( x ) - C .

g ( V D ( g )) W ,

. tg(e

) , .

*. , -- .
Hint. Definition of a compact + Lemma .

(). , , () , () . . () , . , . () () .

* ( ). () , . () , , . () , , , .
, -- , , / .

*. f C ([0, 1]), f (0) = f (1). () , f ( x ) 1/2. () 1/2 ?


(). () f (. e. f ( x ) = f ( y ) x = y ) , f . () f : [a, b] [a, b], f (a) f (b). . () . . . . () . . . . ( f (a - 0), f (a)) . (). , - , . ( ). f C ([a, b]), f [a, b]. f
-1

- . .
( .) . , f ( x ) = o( g( x )) x a, a f ( x ) = ( x ) · g( x ), lim ( x ) = 0. , f ( x ) = o(1) x a , lim f ( x ) = 0.
x a x a

, o- , x a .

( ). ) x = o(1) x 0; ) x = o(1) x 57; ) x 2 = o( x ) x 0; ) x = o( x 2 ) x ;

: [ f (a), f (b)] [a, b],

[ f (a), f (b)]. . . () () , f [ f (a), f (b)], () () f -1 ( x ) . . , , .

( o-). , () o(1) + o(1) = o(1); () o(57 f ) = o( f ); () o(1) - o(1) = o(1); () o(1) · o(1) = o(1); () o( x ) + o( x 2 ) = o( x 2 ) x ; () h( x ) a, x a , f ( x ) = o( g ( x )) f ( x ) · h ( x ) = o( g ( x )). . ln(1 + x ) = x + o( x ) x 0. , lim
x 0

ln(1 + x ) = lim ln(1 + x ) x x 0

1/ x

= ln lim (1 + x )
x 0

1/ x

= ln(e) = 1,

q.e.d. ( ). , e x = 1 + + x + o( x ) x 0. . f a (a D ( f ), a -- D ( f )), a f ( x ) - f (a) A f a f (a) (
df df ( a) dx dx
x =a

f ( x ) - f ( a ) = A · ( x - a ) + o( x - a ), A . ).

()


. () , A f ( x ) - f (a ) . , A = lim x -a x a

g(a) = 0, a, g
f g


f

() , f a
h0

, lim

f (a + h) - f (a ) . h

( a) =

f (a ) g (a ) - f (a ) g (a ) . g 2 (a )

. . ( f · g )( a + h ) - ( f · g )( a ) = f ( a + h ) g ( a + h ) - f ( a ) g ( a ) = = ( f ( a ) + f ( a ) h + o( h ))( g ( a ) + g ( a ) h + o( h )) - f ( a ) g ( a ) = (. ). . () . () . () ? () . : () -3 x 3 - 2 ; () 1,5e2 x - 2 x -3 sin x ; () tg(/4 - x ) + logx (2).
x +1

o- .

. () 57 = 0; () x = 1; () ( x 2 ) = 2 x . , () sin ( x ) = cos( x ). ,
h0


= ( f ( a ) g ( a ) + f ( a ) g ( a )) h + o( h )

( x + h )2 - x 2 = (2 x ) h + h 2 = (2 x ) h + o( h ).

lim

2 sin(h/2) cos( x + h/2) sin( x + h) - sin x = lim = h h h0

= lim


h0

s in ( h / 2 ) cos( x + h/2) = cos x . h /2

() ln ( x ) = 1/ x . ,

,

ln( x + h) - ln( x ) = ln(1 + h/ x ) = h/ x + o(h/ x ) = (1/ x )h + o(h). . () : (c · f ) (a) = c · f (a). () . () sin 1 , x sin 1 , x 2 sin 1 x x x 0, . () :
d f ( a x + b) dx
x=x

. ( ( ( ( ) ) ) ) f f f f

,





; ; , ; , f .

=a
0

df dx

x =a x0 +b

.
x

( ). f a, g b = f (a), g f a, ( g f ) (a) = g (b) · f (a). . ( g f )( a + h ) - ( g f )( a ) = g ( f ( a + h )) - g ( f ( a )) = g ( f ( a ) + t ) - g ( f ( a )),

. a, a. . , ( , a -- D ( f )). ( ). f g a (a -- D ( f ) D ( g)), a, ( f + g ) ( a ) = f ( a ) + g ( a ), ( f g ) ( a ) = f ( a ) · g ( a ) + f ( a ) · g ( a ). . ().

() sin(5 - 7 x ), loga ( x ), a .

t = f (a + h) - f (a). t 0 h 0, g b = f (a) , , t = f (a + h) - f (a) = f (a)h + o(h), o(t ) = o(h), ( g f )(a + h) - ( g f )(a) = g (b) f (a)h + o(h), ( g f )(a + h) - ( g f )(a) = g(b + t ) - g(b) = g (b)t + o(t ).

. : () x ; () ln2 ( 3 x - 1); () x x ; () log x ( x ).


( ). f : X g : Y X (. e. g = f -1 ). f a, f (a) = 0, g b = f (a), g b = f (a g (b) = 1/ f (a). . .

Y ), -

() (a)
.

F ( x ) = f ( x )( g(b) - g(a)) - g( x )( f (b) - f (a)).

. () . () « f (a) = 0» « g b = f (a)»? () . (). f : U (a) a, f a ( ) , f (a) = 0. . a -- ,
f (a + h) - f (a ) h f (a + h) - f (a ) h

0 h > 0 0 h < 0.



: f (a) = 0.

( ). f : [a, b] [a, b] (a, b), : (a) () f (a) = f (b) (a, b) f () = 0; () (, )
(a, b) f (b) - f (a) = f () · (b - a).

-

() () f g . (a, b) , g ()( f (b) - f (a)) = f ()( g(b) - g(a)). . () f x M , xm [a, b]. , f const; f = 0 . () (a) F ( x ) = f ( x )(b - a) - x ( f (b) - f (a)).
( , ) a , f ( x ) ( f ( x ) f ( a )). f ( a)

. () . « »? () , x [a, b] f ( x ) = 0, f = const [a, b]. () M , ( f 0 M ) f = const M ? () , , V , T , V T . () , ( ) , V , T , V T . () , , ( ) 10 /2 ? . f -- (a, b), (a) f -- ( ) (a, b) f |(a,b) 0 ( 0); () f |(a,b) > 0 f () (a, b), f |(a,b) 0 f (a, b). f < 0 ( f 0). . . . () , . ) . () , f (a) > 0, a f ? () , f |(a,b) > 0 f [a, b), f () [a, b). () , e x > 1 + x x = 0. () , e x > 1 + x 57 /57! x > 0. () , 2ab ln(b/a) < b2 - a2 0 < a < b. () e e ; () 19005/7 + 995/7 19995/7 . () cos(2005) 1 + cos(2004).
.

. f : U (a) a ° U (a). : + - (a) f |U (a) > 0 f |U (a) > 0 f a; + - () f |U (a) > 0 f |U (a) < 0 f () a;


() f |

(a) f |

+ U (a) + U

< 0 f |

- U (a) - U

(a)

< 0 f |

<0 ...

(a)

>0 ...


.
.) (


. (. ()). . f a f (a) = 0, f (a) > 0 (< 0) a f (). f (a) = 0 . . () . () , ?

. (an ) -- . ()
n

an = a1 + a2 + ... + an + ... , n =1

S n =
k =1

ak = a1 + a2 + ... + an -- n- . n

lim Sn = S,
n =1

an -

a n ). , S -- (S = n =1 . , () (Sn ) ( ) , Sn -- ; () (, , ) , ;


()
n =1

an , lim an = 0, .
n

. : 1 + 1 ; () 1 1 () ; () . n n 7 n =1 5 n=1 n(n + 1)(n + 2) n=1 n(n + 2000) ( ).


an n =1

, > 0 N n > N m 0 an + an+1 + ... + an+m < . . ( x ) : sin(n x ) () ; () sin(n x )? () ). 2n ( ). , , : () ( ); () . () ( .) N 0 an bn , bn -- an -- ( an -- bn -- ). . an , |an |.


. , , . ( ). . () ( .) (an ) an 0, a1 - a2 + a3 - ... ; , an > 0, an 0, ; () ( .) (an ) , bn , an bn . () ( .) (an ) an 0, bn , an bn .
.
n n

ak bk = a
k=1

n+1

Bn -

(a
k=1

k+1

- ak ) Bk .

q < 1 an (), q > 1 q = 1 .

. () ( .) q = lim n |an |,
n

an -- ,
|a |

() ( .) q = lim n+1 , n |an | ... * (- ). () s (s) = () (.) , (s) =
-1 ( 1 - 1s p ). ( , !) () , Re (s) > 1 - . , , ( -- ?) - ( ) Re (s) = 1/2.

1? ns

. , ( ) : () 1 + 1 + 1 + 1 + 1 + ...; () 1 - 1 + 1 - 1 + 1 - ...;
2 4 6 8 3 5 7 9

1 , , ; () n () 1 - 1 - 1 + 1 - 1 - 1 + 1 - 1 - 1 + ...;
2 4 3 6 8 5 10 12

1 1 , a > 0; () 5n sin 1 ; ; () 7n 1 + an n(n + 2000) 2 + (-1)n (-1)n () (-1)n ; ; ( ) n n + (-1)n x n ; () n! x n ; () e x , sin x , cos x ; () n x n ; () n n! ; () 1; () ln(1 + x ); () nn nn n sin(n x ) . () n x n (n -- n- ); () n ( ...). . () . () ( .) (), , ; , . () , , ( ), (...), . . ( ). : () , , ; () , ; () . , . * ( ). , 0 < an < 1, an , () (1 + an ) = ; () (1 - an ) = 0. *. : n () 1 - 1 ; () ; () (1 + x 2 ); 1 - 32 2 n n +1 2 2 + 2 + ... + 2 ; () cos xn ; () ( ) 2 x 2. () () 1- n ()


- . .
( .) . U -- . U C (U ). C (n) (U ) , n U : C
(n)

() , x ?

n

f

(n)

( x ) = 0, f ( x ) -- -

, n. C (U ) (. ) . . C (U )? . n , C (n) , C (n+1) . ( ). U -- a f C (U ). , : () ( ) (. . f ( x ) ( x - a)); () f (a) = f (a) = ... = f (k-1) (a) = 0, f (k) (a) = 0, k = max {n: f ( x ) ( x - a)n }. k a. () a , f ( x ) = 0 x = a ? ( ). f -- , n . () , P , n, n n f . P . () , P n P ( x ) = P ( x0 ) +
P ( x 0 ) P (n ) ( x 0 ) P ( x0 ) ( x - x0 ) + ( x - x 0 )2 + ... + ( x - x0 )n , 1! 2! n!

(U ) = f C (U ) : x U f

(n)

(x) f

(n)

C (U ) .

. U -- a . , f , g C (n) (U ) a, . a a. -- , . . () ; () . . . f ( x ) g( x ) . , ( f + g)( x ). () . () . ( ). . f -- . : f (0) +
f (0) f (0) 2 f (0) 3 x+ x+ x +... 1! 2! 3!

f ( a ) = 0 ( x ) C ( U ) : f ( x ) = ( x - a ) ( x )

x0 -- . () , P , n, f ( x ) - P ( x ) x0 n+ . P . () , n x f (n) ( x ) = 0, f ( x ) -- ?

, () , () , (. . ), () . () , . ( , .) () , , . ( , , . . .) () x0 (a)-- (). ( ). U -- a , f ( x ) C (U (a)) (, ).


() ,
| f ( x )|
| x - a|n M , n!




. :

() lim x ln( x ), ; () lim ( ( ( ( (

M = sup | f (n) ( x )|. () ( .) , x U (a) , a x ,
f (n - 1 ) ( a ) f (a ) ( x - a) + ... + ( x - a) f ( x ) = f ( a) + 1! (n - 1)!
n -1

+

f

(n )

() ( x - a) n . n!

() « ». () ( .) , f a n , : f ( x ) = f ( a) +
f (a ) f (n ) ( a ) ( x - a) + ... + ( x - a ) n + o(( x - a ) n ). 1! n!

exp( x )( x - 1)3 s in x - 1 / 2 ; ; () lim x 0 x 1 ln x sin( x )2 x /6 cos x - 3/2 cos x - exp (- x 2 /2) ; ) lim tg( x ) x - 1/2; () lim x4 x 0 x 1/2+0 exp( x ) sin( x ) - x (1 + x ) 7 7 ) lim x7 + x5 - x7 - x5 ; ; () lim x + x3 x 0 3 2 x -2 ) lim x 3 + x 2 - x - 1 ; () lim ; x 2 1 + 4 x - 3 x -1 x + x + x + 1 1 - (cos x )sin x ; () lim x x ; ) lim ( x - x 2 ln(1 + 1/ x )); () lim x + x3 x +0 x 0 tg(sin x ) - sin(tg x ) . ) (. . ) lim x 0 arctg(arcsin x ) - arcsin(arctg x )

( .) . () sin( x ); () arctg( x : () cos( x ); () tg( x ); () e x ; ); () ln(1 + x ); () (1 + x ) , .

*. : () ; () e; () ln 2. . ( ). () f ( x ), g( x ) [a, b] (a, b), x [a, b] g ( x ) = 0. f (a) = = g(a) = 0 ,
lim
f (x ) f (x ) = A lim = A. g ( x ) x a +0 g ( x )

x a +0

() f (a) = = g(a) = .


- .
( .) f -- , [a, b]. [a, b] a = x0 < x1 < ... < xn-1 < xn = b. ( ) = max { xi - xi-1 }.
i =1,...,n

() , , S( f , ) s( f , ). () , inf S( f , ) sup s( f , ).



-

() , f

( )0

lim (S( f , ) - s( f , )) = 0.

= (1 , ..., n ), i [ xi-1 , xi ] -- , .
n

* ( ). , f :
b

S ( f , , ) =
i =1

f ( i )( x i - x

i -1

).

() f -- inf S( f , ) = sup s( f , ) =

f (x) dx;
a

() sup s( f , ) = inf S( f , ), f . . , , y = 0, x = a, x = b (. . ), . ?
1

. f ( ) [a, b], S = lim S( f , , ), . .
> 0 > 0 ( ) < |S( f , , ) - S| < . S f [a, b]
b ( )0

f ( x ) d x
a [a , b]

f ( x ) d x. , -

. ()
0

x dx;

[a, b], ([a, b]). . () , . () . () , , , . -- [a, b]. Mi =
n n

() y = x 2 [0; 1]; ()
0

cos x d x ; ()
0

sin x d x .

( ). () . () , [a, b] f
n

sup
x [ x
i-1

f ( x ),

, xi ]

mi =

x [ x

inf
i-1

, xi ]

f ( x ).

> 0 > 0 ( ) <

( f , [ x i , x
i =1

i -1

]) ( x i - x

i -1

)<

S( f , ) =
i =1

Mi ( xi - x

i -1

)

s( f , ) =
i =1

mi ( xi - x

i -1

)

. . () , , S( f , ) S( f , ), s( f , ) s( f , ). () ?

f [a, b]. () ? ( ). , f , () [a; b];
f M ( f , M ) = s up | f ( x ) - f ( y ) | .
x, yM

> 0 > 0 x , x [a, b] | x - x | < | f ( x ) - f ( x )| < .


() ( f -- f ); () ; () ; () . () , . ( ). , ([a, b]) -- () , () - .
Hint. f g = 1/4(( f + g)2 - ( f - g)2 ), h

b

() f C [a, b], f ( x ) 0 [a, b] f ( x ) d x = 0, f ( x ) 0 a [a, b]. () f ( x ), b

[a, b],
a

f ( x ) d x = 0,

[a, b]. * ( ). () , [a, b] C ([a, b]) ( ) |b - a|. ()
b

() , f ([a, b]) | f | ([a, b]), ? () , ([a, b]) . g C ([a, b]), () C ([a, b]), () . ()

h2

.

f

f ( x ) g( x ) d x ,
a

( ). () , [a, b] ([a, b]).
b

() f

() . : () f ( x )
b

a

f ( x ) g( x ) d x , g

([a, b]).

. , . ,
b

([a, b]),

( ). b b

f

f ( x ) g( x ) d x
a

g( x )
b

f (x) dx
a a

g( x ) d x ;

()
a

f (x) dx
a

| f ( x )| d x .

() ( .) f, g ([a, b]), m = inf f ( x ),
x [a , b]

M = sup f ( x ),
x [a , b]

C ([a, b]) g ([a, b])? * ( ). f [a, b] . ( , .)

g [a, b], µ [m, M ] :
b a b

( f · g )( x ) d x = µ g ( x ) d x .
a

.

([a, b]) , -


- . --.
( .) (1) ([a, b]) . . , () f ([a, b]) [c, d ] [a, b] f ([c, d ]); () c [a, b], f ([a, c]) f ([c, b]) f ([a, b])
b c b

. f ( ) f ( x ) d x . . () ; () x ; () f (a x + b), f ( x ) d x = F ( x ) + C ; () a x ; () sin x , cos x ; () sh x , ch x . . . f -- [a, b].
x

(x) =
a

f (t ) d t ,

f (x) dx =
a a

f (x) dx +
c

f (x) dx.
(1)

x [a, b].



. () ( -.) F
b a

([a, b]),

, () ( x ) [a, b]; () f x0 [a, b],


F ( x ) d x = F ( x ) , F ( x ) = F (b) - F (a).
a a

b

b def

( x0 ) = f ( x0 ) (. . d dx

x

x0 ,

f (t ) dt = f ( x ) a

. , .



() , f [a, b], f (1) ([a, b])?
2

f ). ? . ( ).
b

. ()
1 b

1 d x ; () x2

57 1

1 d x ; () x +2

/4 0

() f , g

(1)

([a, b]),
a

f ( x ) g( x ) d x = f g - g( x ) f ( x ) d x .
a a

b

b

cos ( x ) d x .

2

,
a

f ( x ) d x , f

() . . () x 2 cos x d x ; () loga x d x ; () arcsin x d x , arccos x d x ; () arcsh x d x , arcch x d x . ( ). () f C ([a, b]), C (1) ([, ]), ([, ]) [a, b], () = a,
b

F , F ( x ) = f ( x ). . F M , x M F ( x ) = f ( x ). ( ?). () , M -- (, ), f M , F1 - F2 C = const. () f ( x ) = 1 x . () , M . ? () ?

F1 F2 -- n -

( ) = b. ,
a

f (x) dx =


f ( ( t )) ( t ) d t .

() : f ( ( x )) d ( x ) = F ( ( x )) + C . . x d x ; () tg x d x , th x d x ; () 1 - x2 dx , dx () arctg x d x , arcth x d x ; () ; 2 1- x 1 + x2


()

dx , 1 - x2

dx ; 1 + x2

()

x dx 1 + x2 + 1 ± x2 dx. (1 + x 2 )
3

.

- . Calculations
( .)
y b
2



*. ()

() *.

x a

2

+

= 1.

P(x ) d x , P ( x ) Q ( x ) -- ? Q(x )

Definition . The moment of inertia of a rigid body with respect to some axis is I = r 2 dm, where dm is the mass of an infinitesimal volume element, i. e. dm = dV for a body with constant density, r is the distance from that element to the axis. Here we'll be dealing with objects of constant density. Problem . (a) Find the moment of inertia of a stick with respect to the axis going through the middle of the stick perpendicularly to its direction (the length of the stick is L and its mass is M ). (b) Calculate the moment of inertia of a solid disk, with respect to the axis going through the center of the disk perpendicularly to its plane (the disk has mass M and radius R). (c) The same as in b), but now with respect to an axis passing through the center of the disk and laying in its plane. (Hint: Think of the disk as of a collection of bars with different lengths.) (d) Repeat b) and c) for a ring of radius R and mass M . (e) (Pythagorean theorem) Prove that for any solid body in the xy plane Iz = I x + I y . (Ik is the mom. of inertia with respect to k-axis) Problem . Now that you feel yourself comfortable with the moments of inertia, tr y to calculate the moments of inertia of: (a) a hollow sphere (with respect to the axis going through the center of the sphere); (b) a solid sphere. (Hint: think of a solid sphere as of a collection of hollow spheres.) Definition . The position of the center of mass for a given rigid body is given by R= dmr dm .

Problem . (a) Find the center of mass for a semi-circle of radius R with uniform constant density. (b) The same for a half-disk of radius R with uniform constant density. Problem . (a) Find the center of mass for a hemisphere of radius R with uniform constant density. (b) Ibid for a half of an ellipsoid with semi-axis a, b and b, cut in half along the a-axis. Problem . In the th grade you learned in your physics course that the oscillation period of a pendulum is T = 2 (l / g). Now let's find a cor-


rection to this formula: using conser vation of energy one writes (1/2)l ( )2 = g(cos - cos 0 ), where 0 is the angle of maximum deflection for the pendulum and = = d /dt . Integrating we get the following formula for the period:

0

probable speed obtained from d

square of the velocity of a molecule in gas v 2 . Compare it to the most
dv dN dv

= 0.

Problem . Find the electric field along the axis of a uniformly charged ring. The ring has charge Q and radius R. Problem . Find the electric field along the axis of a uniformly charged disk. The disk has charge Q and radius R. Problem (hydrogen Atom). Let us find the ground state energy of an electron inside the hydrogen atom. Probability to find a ground state electron inside the atom is p (r ) = |(r )|2 = (1/(a3 ))e-2r /a , where r is the distance between the nucleus and the electron and a 5 â 10-11 m is the Bohr radius. (By the way, do you know what is?) The potential energy of the electron in the electric field of the nucleus is U (r ) = -e2 /(40 r ), where e is the magnitude of the electron's (or proton's) electric charge, 0 is a constant that you might have seen in your Physics class. To find the ground state energy we need to integrate p (r )U (r )/2 over the whole space (1/2 is due to virial theorem, which you do not know yet, but trust me on this one). The final result for the ground state energy is


T = 4 ( l /2 g )
0 2 4

d cos - cos
0

.

Expanding cos = 1 - /2 + /24 (same for 0 ), expanding the inverse 2 square root in the integrand around 0 - 2 to the next-to-lowest order, and integrating over find the correction to the period.
Hint. you may find the substitution = 0 sin useful to do the integral.

Problem (is there any life on Mars?). A spaceship approached Mars and hung over it, being motionless with respect to the planet's axis. At this ver y moment a famous scientist James J. Jones saw little green humanoids on Mars surface. He was dumb-founded: he absent-mindedly opened the hatch and fell out of the spaceship. When would Dr. Jones fall on the innocent creatures ? The radius of Mars is R, the radius of a ship orbit "-- R . The mass of planet Mars is M . Estimate this time.
Hint. Use conser vation of energy to constr uct differential equation which you can turn into an integral. The potential energy of some mass m in graviof the two masses and G is Newton's constant. tational field of mass M is U = -G
mM , where r is the distance between the centers r

E1 =
0

4r 2 1 p (r )U (r ) dr .
2

(1/2)(dr /dt )2 = G M (1/r - 1/ R )

Find this ground state energy by direct integration. Problem . For a comet moving in the gravitational field of the sun the solution of the equations of motion yields:


t=

min

dr


Problem . An ideal gas is adiabatically compressed (expanded) from some volume V1 to V2 . Find the work done by the gas, if , P1 , V1 , V2 are known. Problem . The number of gas molecules having velocity in the inter val (v , v + dv ) is d N = Cv 2 e
-mv 2 /2k T

(2/m)( E - U (r )) - ( L2 /(m2 r 2 )) ( L/ r 2 ) d r (2m)( E - U (r )) - ( L2 /r 2 )

;

=

min

,

dv ,

where m is the mass of a molecule, T is the temperature, k is the Boltzmann constant, C is some constant irrelevant to the problem. Find the average
Truly speaking, they may be not the same he saw. A mean value of variable is, by definition, = of . ( t i ) ( t i )( t ) ( t i )( t )
i i

where E is the energy of the comet, m is its mass, t is the time, ( , ) are the polar coordinates, U (r ) is the potential energy, L is some constant (angular momentum). For a comet with E = 0 coming to Sun from infinity, show that the trajector y is a parabola. The potential energy is U (r ) = - GrM , G is the gravitation (Newton's) constant, M is the mass of the Sun. When E = 0:
min

=

L2 . 2m M G

, where is a density



( ) . . . . . . . . . . . . . . , , . . . . . . ( ). -- ( -- ). . e . e 0,01. e. . . 3 < < 4. . . . . . . . . ( ). . . . . , . . . . . e . . , . . . . (, ). . . , . . , . -- . . . .

. . . . . , . ( ). , . . . f ( x + h) = f ( x ) + Ah + o(h). . , , , . , . . . , , . . . ( ). . , , . . . . . . . . . .


() .

- .
( .) . , . . , () , () , () , () , () -- ? . , , . . , . ( ). () 4 , . , . () n , . , . () n , 1. , n 1. () n . () ( .) X -- n , X 2, I -- . . f : I , , , . . f : I , ( !) . () ,
x1 , x2 I [0; 1] f ( x1 + (1 - ) x2 )
2n , X . n+1

. () , (a, b) . () [a, b]? () , (a, b) ? () , f , . . () , f : (a, b) , x (a, b) :
f± ( x ) = lim

. () , a + b > 2 a512 + b512 > 2; () , a + b > 2 a57 + b57 > 2.

h±0

f ( x + h) - f ( x ) ; h

() () ()

f- ( x ) f+ ( x ). , x < y f+ ( x ) f- ( y ). , , , , .

. , f C (1) ((a, b)), f f . ( ). () , (a, b) f , , f (, ); () f (a, b) , f 0 (a, b). ( ). f : (a, b) , t1 , ..., tn (a, b), 1 , ..., n 0, 1 + ... + n = 1, f (1 t1 + ... + n tn ) 1 f (t1 ) + ... + n f (tn ). -

. () ( .) x1 , ..., xn > 0
n

f ( x1 ) + (1 - ) f ( x2 ). .

() ( --.) x1 y1 + ... + xn yn
Hint. f (t ) = t 2 .
2 2 2 2 ( x1 + ... + xn )( y1 + ... + yn ).

x1 · ... · x

n

( x 1 + ... + x n )/ n .

, .


() ( .) p , q > 0, 1/ p + 1/q = 1, xi , yi ¨ x1 y1 + ... + xn yn (x + ... + x
p 1 p 1/ p n)

0

(

q y1

+...+

q yn )1/q

- .
(
1/ p

( p = q = 2 -- ...). () ( .) p > 1, xi , yi (( x1 + y1 ) + ... + ( xn + yn ) )
p p 1/ p

.)

0
p y1 p + ... + yn )

(x + ... + x

p 1

( p = 2 -- ()). ( ). S ( x1 , ..., xn ) =
x +...+ x n
1 n

p 1/ p n) n

+(

, -

, , . . n i (t ) -- . , . . . = (I ) n . . C - y = | x | -1 x 1? . . , : [, ] [a, b], [, ] 1 (()) = 2 () . [] , , , [] -- . . , , . ? . t () (t ) = (1 (t ), 2 (t ), ..., n (t )). . []
b a b

1/

: I = [a, b]

n

;

t (t ) = (1 (t ), 2 (t ), ..., n (t )),

-- x1 , ..., xn , = 0. , : () S ( x1 , ..., xn ) S ( x1 , ..., xn ) < ; () lim S ( x1 , ..., xn ) = max xi ; lim S ( x1 , ..., xn ) = min xi . () lim S ( x1 , ..., xn ).
0 + i =1,...,n - i =1,...,n

1 : I1 = [a, b]

n



2 : I2 = [, ]

n

|(t )| dt =

( 1 ( t ))2 + ( 2 ( t ))2 + ... + ( n ( t ))2 d t .
a

. () . () , .
, , . .


. () -- : y = f ( x ); () , : = ( )? . () () x 2 = 2 py ( p -- ); () ( ) x 3 = y 2 ; () ( ) , , ( ); () () x 2/3 + y 2/3 = 1; () ( ) = a [0, 2]; () () = 1 + cos ( ?); () = ( + 1/ )/2 [1, 3]. ! ( ). , a b y = c sin( x /b), c = , . ( · , · ) -- n . . , () ( ) a2 - b2 .

, () s, d 1, . e. ( ) () ds , ... d2 (s) ( ) d s2 (s). ! . k (s) = . k () , () , y = f ( x ). . () , () , () , () . . k . . () , () = ae () , () 2 = 2a2 cos 2 . !

m

,

d ( ( t ), ( t )) = ( ( t ), ( t )) + ( ( t ), ( t )); 1 2 2 2 1 dt 1 () |(t )| const t (t ) (t ), ; () , . , C - «». : . [] , () , () t |(t )| = 0.
t

( ). 0

s(t ) = |(t )| dt .
x0 x1 .


- .
( .) n- n + 1 . deg = 1 . deg = 2 -- . . , ( ?) , R -- . . . , () k = 1/ R; () .
. , ( ), .

. [ · , · ]. - v w, 3 -- . . b = [v, n] . ( ?) ( ). : d v = 0 · v + k · n + 0 · b, ds d n = - k · v + 0 · n - · b, ds d b = 0 · v + · n + 0 · b, ds . . () , () . . : () ( ) (t ) = (a cos t , a sin t , bt ); () (t ) = e-t (cos t , sin t , 1); () (t ) = a(ch t , sh t , t ); () (t ) = (t 2 3/2, 2 - t , t 3 ); () ( t ) = (3 t - t 3 , 3 t 2 , 3 t + t 3 ). . *. , k (s) (s), (s) , . *. () n . () . () .
. n â n SO(n) n .

( ). = (s) (s -- ) k , : d v = 0 · v + k · n, ds d n = -k · v + 0 · n; ds

v -- , w -- , n = w -- |w| . . () k 0, () k const. . , k (s), (s) , . . : () x = 3t , y = 3t 2 , z = 2t 3 (0, 0, 0) (3, 3, 2); () x = e-t cos t , y = e-t sin t , z = e-t , t (0, +); () ( y - x )2 = a( y + x ), y 2 - x 2 = 9 z 2 (0, 0, 0) ( x0 , y0 , z0 ).
8

*. , ?


- .
( .) , 1. (), , -- . (n- «» ). n ( ); ( 1/2n ). . ; . , «» ( ) . A Pr[ A]. . , n- () ; () ; () ; () ; () . () 100- : 49 50 ? ? . () n- «» ( 1 6)? : ? () . ? n . () k ? () k ? . N , 1, ..., N , . () ?

() , 1 ? () , 3 2, 1? () , ( i i - )? . 6 1 49, 6 1 49. () ? () ? () 5 ? . A B Pr[ A | B] = Pr[ A B]/ Pr[ B]. (, Pr[ B] > 0.) . , : () ; () ; () . . A B, Pr [ A | B ] = Pr [ A ] . . , : A B, B A. . () « » « , 3» ? () ( ) « » « » ? () « » « ». () « » « ». . B C A. , « B C » A? . n . A k , B -- n - k . , A B . . Pr[ A| B], Pr[ B| A], Pr[ A] Pr[ B]. ( .)


*. N N , . . 0, 1, 2, ... ( N )? *. n n . , , , ?

¨ 57

. . . 60 â 90 / . . . . . . . , , ., . . ( )

­

­

.

« », ., . . . ( ) ­ ­ . E-mail: biblio@mccme.ru