ЛЕКЦІЇ ² ВПРАВИ

ñòåïåíåâîãî ðÿäó. Ïðè x =±R ðÿä ìîæå àáî çá³ãàòèñÿ, àáî
ðîçá³ãàòèñÿ.
12.5.3. Âèçíà÷åííÿ ðàä³óñà çá³æíîñò³ ñòåïåíåâîãî
ðÿäó
Íàâåäåìî ñïîñ³á âèçíà÷åííÿ ðàä³óñà çá³æíîñò³ ñòåïåíåâîãî
ðÿäó (12.5.1). Ðîçãëÿíåìî ðÿä
a
n+
1
lim 0
n→∞
an
≠
1
. Ïîçíà÷èìî éîãî ÷åðåç
R . Òîä³
a x a x
lim = =
n→∞
R
.
n+
1
n+ 1 n+
1
x lim
n
ax
n→∞
n
an
∞
n
∑ ax
n . Íåõàé
n=
0
Ïðè êîæíîìó çíà÷åíí³ x ñòåïåíåâèé ðÿä ñòຠ÷èñëîâèì
ðÿäîì. Òîìó çà îçíàêîþ Äàëàìáåðà ðÿä
∞
n
∑ ax
n çá³ãàºòüñÿ,
n=
0
x
ÿêùî 1
R < , òîáòî ïðè |x| 1
n
n→∞
n
ax
n
R
³ çàãàëüíèé ÷ëåí ax íå ïðÿìóº äî íóëÿ
n
ïðè n →∞.
n
Òàêèì ÷èíîì, ñòåïåíåâèé ðÿä ∑ ax
n çá³ãàºòüñÿ âñåðåäèí³
³íòåðâàëó (–R, R) ³ ðîçá³ãàºòüñÿ ïîçà íüîãî, òîáòî ðàä³óñ
n=
0
çá³æíîñò³ ðÿäó
∞
a
R = lim
n
n→∞
a
. (12.5.6)
Çàóâàæåííÿ. ßêùî ãðàíèöÿ â (12.5.6) äîð³âíþº íóëþ,
òî ñòåïåíåâèé ðÿä çá³ãàºòüñÿ ò³ëüêè ïðè x = 0, òîáòî R =0.
Ïðè R = ∞ ñòåïåíåâèé ðÿä çá³ãàºòüñÿ íà âñ³é ÷èñëîâ³é ïðÿì³é.
n+
1
Ïðèêëàä 12.5.1. Çíàéòè ðàä³óñ çá³æíîñò³ ðÿäó
x
2
x
n
x ∞
n
x
n=
0
1+ + + K+ + K = ∑ .
1! 2! n ! n !
1 1
= , a =
! 1 !
. Îòæå,
Ð î ç â ’ ÿ ç à í í ÿ. Òóò n
n+
1
n ( n+
)
a
( n + )
1!
R = lim = lim( n+ 1)
=∞.
n→∞
n!
n→∞
Âèõ³äíèé ðÿä çá³ãàºòüñÿ àáñîëþòíî äëÿ ∈−∞∞ ( , )
x .
Ïðèêëàä 12.5.2. Âèçíà÷èòè ðàä³óñ çá³æíîñò³ ðÿäó
∞
n
x
∑
n .
n=
1
1 1
Ð î ç â ’ ÿ ç à í í ÿ. Òóò an
= , an+
1
= . Òîìó
n n + 1
n + 1 ⎛ 1⎞
R = lim = lim ⎜1+ ⎟=
1 .
n→∞
n n→∞⎝
n⎠
Ðÿä çá³ãàºòüñÿ àáñîëþòíî äëÿ x ∈− ( 1,1)
.
Ïðèêëàä 12.5.3. Âèçíà÷èòè ðàä³óñ çá³æíîñò³ ðÿäó
∞
∑ nx ! n
.
n=
0
Ð î ç â ’ ÿ ç à í í ÿ. Òóò a n a ( n )
n
( n )
= !, = + 1 ! ³
n+
1
n! 1
R = lim = lim = 0
n→∞
+ 1! n→∞
n+ 1
.
Ðÿä ðîçá³ãàºòüñÿ íà âñ³é ÷èñëîâ³é ïðÿì³é, êð³ì òî÷êè
x =0.
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