ЛЕКЦІЇ ² ВПРАВИ

Ïðè a = 0 ðÿä Òåéëîðà º ñòåïåíåâèé ðÿä â³äíîñíî íåçàëåæíî¿
çì³ííî¿ x:
( 0)
( ) ( )
( n ) ( )
f′ 0 f′′
0 f 0
+ + + K+ + K, (12.5.10)
1! 2! n!
2
n
f x x x
ÿêèé íàçèâàºòüñÿ ðÿäîì Ìàêëîðåíà.
Äëÿ ðîçêëàäàííÿ äàíî¿ ôóíêö³¿ â ðÿä Òåéëîðà ïîòð³áíî:
1) çàïèñàòè ðÿä Òåéëîðà äëÿ äàíî¿ ôóíêö³¿, òîáòî îá÷èñëèòè
çíà÷åííÿ ö³º¿ ôóíêö³¿ ³ ¿¿ ïîõ³äíèõ ïðè x = a ³ ï³äñòàâèòè
¿õ äî çàãàëüíîãî âèðàçó ðÿäó Òåéëîðà (12.5.9);
2) äîñë³äèòè çàëèøêîâèé ÷ëåí R n ôîðìóëè Òåéëîðà äëÿ
äàíî¿ ôóíêö³¿ ³ âèçíà÷èòè ñóêóïí³ñòü çíà÷åíü x, ïðè ÿêèõ
îòðèìàíèé ðÿä çá³ãàºòüñÿ äî äàíî¿ ôóíêö³¿ (òîáòî ïðè
ÿêèõ x lim Rn
= 0 ).
n→∞
Ïðè ðîçêëàäàíí³ áàãàòüîõ ôóíêö³é â ðÿä Òåéëîðà ìîæíà
çàì³ñòü äîñë³äæåííÿ â³äïîâ³äíîãî çàëèøêîâîãî ÷ëåíà R n , ùî
â áàãàòüîõ âèïàäêàõ äóæå âàæêî, äîñë³äèòè çá³æí³ñòü ñàìîãî
ðÿäó Òåéëîðà, ÿê çâè÷àéíîãî ñòåïåíåâîãî ðÿäó.
Ðîçãëÿíåìî ðîçêëàäàííÿ äåÿêèõ åëåìåíòàðíèõ ôóíêö³é â
ðÿä Ìàêëîðåíà.
Ðîçêëàäàííÿ ôóíêö³¿ f(x) =e x . Ìàºìî:
( n
′( ) ′′( )
)
( )
çâ³äêè ïðè x = 0 îäåðæóºìî: ′( ) ′′( )
f x = f x = = f x = e
x
K ,
( n )
( )
f 0 = f 0 = K = f 0 = 1 (äèâ.
ïðèêë. 7.9.7). Çà ôîðìóëîþ (12.5.10) äëÿ ôóíêö³¿ e x ñêëàäåìî
ðÿä Ìàêëîðåíà:
x
2
x
n
x ∞
n
x
n=
0
x
e = 1+ + + K+ + K = ∑ . (12.5.11)
1! 2! n ! n !
Çíàéäåìî ³íòåðâàë çá³æíîñò³ ðÿäó (12.5.11)
( n + )
a
1!
R = lim = lim = lim( n+ 1)
=∞ .
n
n an+
1
n n!
n Îòæå, ðÿä (12.5.11) àáñîëþòíî çá³ãàºòüñÿ íà âñ³é ÷èñëîâ³é
ïðÿì³é.
Ðîçêëàäàííÿ ôóíêö³¿ f(x) = sin x. Ìàºìî:
f′ x = x = ⎛ x +
π ⎞
( ) cos sin ⎜
2
⎟
⎝ ⎠ , ( )
= sin
⎛
⎜x+
n π ⎞
⎟
⎝ 2 ⎠
⎛ π ⎞
( n
f′′ x =− sin x = sin ⎜x + 2 , , f ) ( x)
=
2
⎟ K
⎝ ⎠
. Çâ³äêè, ïîêëàâøè x = 0, îäåðæóºìî: f ( 0 ) = 0,
( ) ( ) ( )
( 4 ) ( )
f′ 0 = 1, f′′ 0 = 0, f′′′
0 = − 1, f 0 = 0, K (äèâ. ïðèêë. 7.9.8).
Ñêëàäåìî çà ôîðìóëîþ (12.4.10) äëÿ ôóíêö³¿ sin x ðÿä Ìàêëîðåíà:
n−
( − ) x
( n−
)
n−
( − ) x
( n−
)
3 5 7
1 2n−1 1 2n−1
x x x 1 ∞ 1
3! 5! 7! 2 1 ! n=
1 2 1 !
sin x = x− + − + K + + K = ∑
.
Ëåãêî ïåðåâ³ðèòè, ùî îòðèìàíèé ðÿä çá³ãàºòüñÿ àáñîëþòíî
íà âñ³é ÷èñëîâ³é ïðÿì³é:
( n + )
( n−
)
a 2 1 !
R = lim = lim = lim 2n( n+ 1)
=∞.
n
n an+
1
n 2 1 ! n Ðîçêëàäàííÿ ôóíêö³¿ f(x) = cos x. Àíàëîã³÷íî ïîïåðåäíüîìó
ìîæíà îòðèìàòè ðîçêëàäàííÿ ôóíêö³¿ cos x ó ðÿä
Ìàêëîðåíà, ÿêå ñïðàâåäëèâå ïðè áóäü-ÿêîìó x. Îäíàê ùå
ïðîñò³øå ðîçêëàäàííÿ cos x îòðèìóºòüñÿ ïðè âèêîðèñòàíí³
âëàñòèâîñò³ ïî÷ëåííîãî äèôåðåíö³þâàííÿ ñòåïåíåâîãî ðÿäó,
â äàíîìó âèïàäêó ðÿäó äëÿ sin x:
3
′
5
′
7
′
n−1 ′
2n−1
′ ′ ⎛ x ⎞ ⎛ x ⎞ ⎛ x ⎞ ⎛( −1)
x ⎞
cos x = ( sin x) = ( x)
− ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ + + +
3! 5! 7! K ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎜ ( 2n
−1 )!
⎟
K,
⎝ ⎠
çâ³äêè
n
( − ) x
( n)
n
( − ) x
( n)
1 1
K K .
2 4 6
2n
2n
x x x
∞
cos x = 1− + − + + + = ∑
2! 4! 6! 2 ! n=
0 2 !
470 471