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[НЕСОРТИРОВАННОЕ]

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Вступ до аналізу. Ч. 1

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* {xn } ! + . ) + ,									lim xn = inf {xn}. %:
									n → ∞
b = lim xn	= lim xn+1 . #
n → ∞	n → ∞
b = lim	x	=	lim	a	x	=	lim	a	lim x = 0 b ,


n → ∞	n +1		n → ∞ n + 1		n		n → ∞ n + 1 n → ∞ n

b = 0 . # " a > 0 * (12.1) . ) *

an	≤	\| a \|n	,
n!	≤	n!	,
n!		n!

* (12.1) a . $" * {xn } ,

			1 n
xn	= 1	+		,

			n

+, " * ! +, " , - ' ! + . & ' ! .*', -:

x = 1 + C1

2 1

+ C

+ ... + C k

+ ... +

= 1 + ∑ C k

n n2

n n

n nk

k =1

n nk

C k =

n(n −1)L(n − k + 1)

(k =

)

1, n

k !(n − k )!

k !

– ! * ,. $":

C k

n(n −1)L(n − (k −1))

−

L 1

−

k !


n	1		1		2		k − 1
xn = 1 + ∑		1 −		1 −		L 1 −		,

k =1 k !			n		n		n

n +1 1

k − 1

xn+1 = 1 +

∑

−

L 1

−

k =1 k !

n + 1

1 −

< 1 −

,...,1 −

k −1

< 1 −

k −1

n + 1

k− 1

(12.2)

(12.3)

# + (12.2) 2

(12.3). , (12.3) ( ( ( -

, k = n + 1) " ' (12.2), ( (. ) +

xn < xn +1 ,

! * {xn } '.

, *

1 −

< 1

(m =

1, n −1

2k −1

k !

1 −

n 1

xn < 1 + ∑

< 1 + ∑

= 1 +

= 1

+ 2 1 −

= 3 −

< 3

−1

3n−1

k =1 k !

k =1 2k

1 −

( " 2 n -

). # * {xn } ! + . & ,

'. # , lim xn . 4" " , * " ! ' e

n → ∞

( ’" * 8 :().

		1 n
lim 1	+		= e .	(12.4)

n→ ∞		n

? e , * " /. 4 , + + *

". & *, + ! + , * " -

" !. @ " '

15 " :

e = 2, 718281828459045... .

? ' * " ! + ' ': e ≈ 2,72 .

8 a > 0 ' e , * " (-

, * " ln a . # !:

ln a = loge a .

. * + 2 ", !-

* " * - 2, + 2 ' ', * ". 6 e

’" + ', " , * " (

! +).

. % * 1, 2 ,..., n ,... , n = [an , bn ] , - , * " , " :

1) n : n +1 n ;
2) lim (bn − an ) = 0 .
n → ∞
# :
a1 ≤ a2 ≤ K ≤ an ≤ an +1 ≤ K ≤ bn +1 ≤ bn ≤ K ≤ b2 ≤ b1 .	(12.5)

" '. ) & ,

, .

. 6 '+ ( (12.5) ,,

n, m : an ≤ bm .

# + {an } ! + , + , sup{an } = c , :

n : an ≤ c ≤ bn ,

( ,, c + * { n } .

% + , c ,. %, ' *

c1, c2 (c2 > c1) , + * { n } . # n

bn − an ≥ c2 − c1 > 0 , * 2) . # .

#. {an }, {bn } , :

lim an	= lim bn = c = sup{an} = inf{bn} .
n → ∞	n → ∞

4 ( * (.

13. . &.

. ( * {xn } . $" ' * -

* :

n1, n2 ,..., nk ,... (nk < nk +1 ) .

. % * {xnk } , * " -

{xn } .

. .( xn = n , ! {xn } – * * .

& * xn = 2k − 1 (k = 1, 2,...) *
	k
, " " " ". + * x′ = 2k
	nk
(k =1, 2,...) * , " + " " ".
% {xn }, {xn′	} , " {xn } .
k	k

6 +, * {xn } - + + {x1, x2 ,..., xn ,...} . % " " -

, ! , " ( {xn } . . -

, * {5,3,9, 7,13,11,...} , ' -

* , +, " ( {xnk } , + *-

&{xnk } k ,

xn1 , xn2 ,..., xnk ,... , nk – * ( {xn } .

. ( {xnk } – * {xn } , ( , - ( ! ( lim xn = a . # a , * "

k → ∞	k
k → ∞
{xn } .
	44

1. $" * xn = (−1)n−1 . # ! * 1, −1,1, −1,... .

& , : −1, −1, −1,... 1,1,1,.. . % 2 , - ', " ', −1, , ', " ', 1. ) + −1 1 –

xn = (−1)n−1 .

2. $" * xn = (1 + (−1)n )n . # ! * 0, 4, 0,

8, 0, 12,… & , : 0, 0, 0,… 4, 8, 12,… . % 2

, ', " ', 0, , ', " ', + ∞ . )-

+ 0 + ∞ – x = (1 + (−1)n )n .
n
. .(! 2		{xn } -

, * " ! {xn } , * "		lim xn . .(-
n → ∞

2 * {xn } , * " -

{xn } , * " lim xn .

n → ∞

. $" * 1,2,3,1,2,3,1,2,3,… . &

! + : 1,1,1,… ; 2,2,2,… ; 3,3,3,… . % 2 , -

', " ', 1, – ', " ', 2, " – ', "

', 3. ) + lim xn
	= 1, lim xn = 3 .
n → ∞	n → ∞

( ) &-* +1. 0

. .( {xn } – ! + *. # , ( -

[a,b] , n : xn [a, b] . $!’, [a,b] -

' d . # ( [a, d ] ! [d ,b] -

* * {xn } . % ( "

[a1, b1 ] . @ + ', b1 − a1 = (b − a) 2 . & [a1, b1 ] + -

, ! (, * -

* * . % ( " [a2 , b2 ] . @ + -

', b2 − a2 = (b1 − a1)2 = (b − a) 22 .

% + ' ( ! +, , *

{[an , bn ]} ,

1) [a1, b1 ] [a2 , b2 ] K [an , bn ] [an +1, bn +1 ] K ;

2) lim (b		− a		) = lim	b − a	= 0 .
			n
n → ∞	n			n → ∞	2n

1 * (1781–1848) – * ( . & (, 2 (1815–1897) – ( * (

) + {[an , bn ]} – "+ * , ' , ,

c , + * [an , bn ] ( n = 1, 2,...). % +,

( * " * {xn	} {xn	} , lim xn		= c .
k		k → ∞	k
		k → ∞

) * [a1, b1 ] * * * -

{xn } , n1 : xn1 [a1,b1]. & [a2 , b2 ] + * -

* * , + n2 : xn2 [a2 ,b2 ] . 0 -

k nk : xnk [ak ,bk ] . # , * {xnk } -

{xn } , k : ak							≤ xn ≤ bk . 6 2 ! ak ≤ c ≤ bk ,
							k
						= (b − a) 2k . ) * {(b − a) 2k } –
	x	− c	≤ b	− a	k	= (b − a) 2k . ) * {(b − a) 2k } –
	n		k		k
	k
* ( . . 10, 1), lim xn = c .
							k → ∞	k
							k → ∞

8 *-& (, 2 .

14. ' ' + ! .

. % * {xn } , * " (, "

ε > 0 N n, m > N : | xn − xm | < ε . ! + ' :

ε > 0 N n > N , p : | xn + p − xn | < ε .

. .( xn = 1 n . % +, " * , *-

'. 6 * ε > 0 . $":

n − n − p

| x

− x | =

−

p .

n+ p

n + p n

n(n + p)

n n

n 1

&, ! 1 n < ε . # n > 1 ε , ,, N + ": N = [1 ε] + 1. # * *.

5 * {xn } *, ! +. (,

" * ,: ε > 0 N n, m > N : | xn − xm | < ε . %: M = max (| x1 |,| x2 |,...,| xN +1 | ) . # n > N ":

| xn | = | xn − xN +1 + xN +1 | < | xn − xN +1 | + | xN +1 | ≤ M + ε .

) + n > N : | xn | < M + ε , ! * {xn } ! +.

)! + ", , , ! ! + -

, * *. % + ! -

* * xn = (−1)n .

" ( 1 *2). , & {xn } -

, ! , & (.

. .! *. .( * {xn } ! +, ! ,

( lim xn = a . 6 "

n → ∞

ε > 0 N n > N : \| x − a \| <			ε	.
ε > 0 N n > N : \| x − a \| <				.
		n	2
$" n, m > N :			2
$" n, m > N :
\| x − x \| = \| x − a + a − x \| ≤ \| x − a \| +\| x − a \| <						ε	+	ε	= ε ,
\| x − x \| = \| x − a + a − x \| ≤ \| x − a \| +\| x − a \| <							+		= ε ,
n m	n	m	n		m	2	2
						2	2

! * {xn } *.

*. .( * {xn } *. #

ε > 0 N n, m > N : | xn − xm | < ε . 2

> * * , ! + ', ' *- & (, 2 + ! + * {xnk } . .(

lim xn		= a . % +, ( lim xn = a . / ,:
k → ∞	k	n → ∞

ε > 0 K

k > K

x − a

%: N1 = max (N , nK0

). # " nk

> N1 , n > N1 -

* {xn } ! :

− x

) + n > N1 :

| x − a | =

x − x + x − a

≤

x − x

x − a

= ε ,

lim xn = a .

n → ∞

# .

. , * {xn } ,

x = 1 +

+ ... +

! +.

% +, ε > 0 N n, m > N :

| xn − xm | ≥ ε . 6 *

n = 2k, m = k . #

| x

− x

|=| x

− x | =

+ ... +

≥

+ 1 k + 2

2k 2k 2

2 2 ) ' 8 (1789–1857) – ( * ( .

# " ε + " 1 2 , + * , -

* ', ' 2 ! +.

15. , &, ! , &.

, & ,.

& ' 2 ", 2 , " ’" + ! ' , " * - 2. # ( ,’" + , * " ' " "

. . r + , * " ': S = πr 2 . 6 '' r , ! ' , +

+ , S , ,' ( r . 6

* "+ ", , " ' m M ,- ' * + ! ', + , * " ':

F = γ mM , r 2

r – * + , γ – * ( . 6- 2 ' m M , + ' * r , -

! ' * F , ! + , F ,

,' r .

" .

. 5 + x " + X -

* * y -

+ Y , + *, + X ( y = f (x) .

6 x , * " , ! . 6 y

, * " , ! (. % f -

, " ( ( , " ', * " *, !

, " ! x , !

" y .

/ + X , * " , * " D f . / + Y , * " , * " E f . . -

, " y = 1 − x2 ! ' " , [−1;1] ,

+ ' * [0;1] . * " x

+ ( + " y ,
" x = −1, x = 1	, ( +

" y = 0 . " ! ", -

', ( + ' , *

" . A 2 ,

( , " ' * " ( !-

) .

$" + + "

" , ! ' " " , + :

xn = f (n), n = 1, 2, 3,...

> ' + !.

1. ( !. 4 ! " '

( ):

y = f ( x) , ! F (x, y) = 0 .

., y = x2 , x (−∞, + ∞) – ( ! "

, " + ( ' x * -

* " x2 .

( ! , " , .

" * " *, * + *

. – * , ! * !-

" " + " * .

2. # ! ( !. 4 ! " ' !, "

* " " ! " :

x	x1	x2 … xn

y	y1	y2		…	yn

	#	yk	= f ( xk ) (k = 1, 2,..., n) .
	., " y = x2 " ! " + ( ":

x	0	0,1		0,2		0,3	0,4	0,5

y	0	0,01		0,04		0,09	0,16	0,25

	4 ( ! ' * , -
' * *							x1 , x2 ,..., xn * 2 "

" * " y1 , y2 ,..., yn . , ! (

! 2 , * " ’" -

' ’', ' ’' , " -

(! *). 6 2 * «? -

!» &./. .

4 ( ! + , . * " ,

! ! ' " ( + , !)

* (! ( ' * ! "), * , +

, + ! ( "

" * , " " * ! ( !

!). %, " , " " ,

+ ! + ( " ' -

. 0 , " * .

3. ( !. 4 ! " ' (, ! + ( , "

(x, y) " * "' * * y = f (x) . 6 + , "

', + " , * , * (,

+ . . , y = x2 , - " ( !):

$ . 16.

! 2 ' * " + ", ’"

" !, " ! ("

), (" , ), -

(" ", ’" " * ' - "), (" ") . (

!, , , ), " ' *

, ' * ', " ' * !

, 2 .

! * ", 2 , * ( , -

' * ( ! ( !). / ' , + -

" ! , " ( , * ". $

( ! , : *, 2 ! + " "

" " .

4. ( !. 0 ' * , " , ! +

', + + ! . " -

( + ' .

, " :

1, & x − (x ),

D (x ) =

0, & x − ( x / ).

, ! +, * - , * " " y = 1, " ' * * !, "

y = 0 , " ' * * !. > ' , +, 50

+, + . ! " * , + + -

, * ! * 2 " , +

+ " ! * " -

>, " *2 ! , ' * " -

" (, " ' * " -

(. " * " .

1., ! .

4 " y = C = const (! + ( "

( , ( + "). )! ' " D f , - + (− ∞ , + ∞) , + * E f , * " * ,

y = C . , " ", " * ! -

* y = C ( . 17).

$ . 17.

2. . 4 " y = xα , α .

)! * " + * + * α . . -

" ( . 1) α = 1 ; y = x ; D f = E f = (− ∞ , + ∞) .

$ . 18.

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