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íà çàâæäè íå ïîðîæíÿ, òîìó ùî áóäü-ÿêèé ñòåïåíåâèé ðÿä<br />
çá³ãàºòüñÿ ïðè x =0.<br />
Î÷åâèäíî, ùî ÷àñòèííà ñóìà ñòåïåíåâîãî ðÿäó<br />
n<br />
Sn( x) = a0 + a1x+ K + anx<br />
º ôóíêö³ºþ çì³ííî¿ x. Òîìó ³ ñóìà<br />
ðÿäó S òàêîæ º ôóíêö³ºþ çì³ííî¿ x, ÿêà âèçíà÷åíà â îáëàñò³<br />
çá³æíîñò³ ðÿäó:<br />
∞<br />
n<br />
( ∑ n )<br />
∞<br />
n<br />
( ) ∑<br />
( )<br />
S = S x = a x àáî f x = a x .<br />
n<br />
n= 0 n=<br />
0<br />
12.5.2. ²íòåðâàë çá³æíîñò³ ñòåïåíåâîãî ðÿäó<br />
Äîâåäåìî òåîðåìó, ùî ìຠâàæëèâå çíà÷åííÿ â òåî𳿠ñòåïåíåâèõ<br />
ðÿä³â. Âîíà ñòîñóºòüñÿ îáëàñò³ çá³æíîñò³ ñòåïåíåâîãî<br />
ðÿäó.<br />
Òåîðåìà 12.5.1 (Àáåëÿ ïðî çá³æí³ñòü ñòåïåíåâîãî<br />
ðÿäó):<br />
1) ÿêùî ñòåïåíåâèé ðÿä (12.5.1) çá³ãàºòüñÿ ïðè<br />
x = x 0 (x 0 ≠ 0), òî â³í çá³ãàºòüñÿ ³ ïðèòîìó àáñîëþòíî äëÿ<br />
âñ³õ x, ÿê³ çàäîâîëüíÿþòü âèìîãó x ≤ x0<br />
;<br />
2) ÿêùî ðÿä (11.12) ðîçá³ãàºòüñÿ ïðè x = x 1 , òî â³í ðîçá³ãàºòüñÿ<br />
äëÿ óñ³õ x, ÿê³ çàäîâîëüíÿþòü âèìîãó x > x1<br />
.<br />
Äîâåäåííÿ. 1) Îñê³ëüêè çà óìîâîþ ÷èñëîâèé ðÿä<br />
∞<br />
∑ n<br />
ax<br />
n 0 çá³ãàºòüñÿ, òî éîãî çàãàëüíèé ÷ëåí n<br />
ax → 0 ïðè<br />
0<br />
n=<br />
0<br />
n<br />
n →∞, çâ³äêè âèïëèâàº, ùî ïîñë³äîâí³ñòü { n 0 }<br />
ax º îáìåæåíîþ,<br />
òîáòî ³ñíóº ÷èñëî M > 0 òàêå, ùî<br />
ax<br />
n<br />
n 0<br />
Ïåðåïèøåìî ðÿä (12.5.1) ó âèãëÿä³<br />
2<br />
x 2 x n x<br />
0 1 0 2 0 n 0<br />
x0 x0 x0<br />
n<br />
< M. (12.5.2)<br />
⎛ ⎞ ⎛ ⎞ ⎛ ⎞<br />
a + a x ⎜ ⎟+ a x ⎜ ⎟ + K+ a x ⎜ ⎟ + K. (12.5.3)<br />
⎝ ⎠ ⎝ ⎠ ⎝ ⎠<br />
³ ðîçãëÿíåìî ðÿä, ñêëàäåíèé ç àáñîëþòíèõ âåëè÷èí éîãî<br />
÷ëåí³â:<br />
n<br />
2<br />
n<br />
0 1 0 2 0 n 0<br />
x0 x0 x0<br />
2<br />
x x x<br />
a + a x + a x + K+ a x + K. (12.5.4)<br />
×ëåíè ðÿäó (12.5.4) íà ï³äñòàâ³ íåð³âíîñò³ (12.5.2) ìåíø³<br />
ïðîòè â³äïîâ³äíèõ ÷ëåí³â ðÿäó<br />
2<br />
x x x<br />
M + M + M + K+ M + K. (12.5.5)<br />
x x x<br />
0 0 0<br />
Ïðè x < x0<br />
ðÿä (12.5.5) ÿâëÿº ñîáîþ ãåîìåòðè÷íó ïðîãðåñ³þ<br />
³ç çíàìåííèêîì q =<br />
x<br />
x0<br />
< 1 ³, îòæå, çá³ãàºòüñÿ. Îñê³ëüêè<br />
÷ëåíè ðÿäó (12.5.4) ìåíø³ ïðîòè â³äïîâ³äíèõ ÷ëåí³â<br />
ðÿäó (12.5.5), òî çà îçíàêîþ ïîð³âíÿííÿ ðÿä (12.5.4) òàêîæ<br />
çá³ãàºòüñÿ, à öå îçíà÷àº, ùî ðÿä (12.5.1) ïðè x < x0<br />
çá³ãà-<br />
ºòüñÿ àáñîëþòíî.<br />
2) Äîâåäåìî òåïåð äðóãó ÷àñòèíó òåîðåìè. Çà óìîâîþ â<br />
òî÷ö³ x 1 ðÿä (12.5.1) ðîçá³ãàºòüñÿ. Ïîòð³áíî ïîêàçàòè, ùî<br />
â³í ðîçá³ãàºòüñÿ äëÿ óñ³õ x, ÿê³ çàäîâîëüíÿþòü âèìîãó<br />
x > x 1 . Ïðèïóñòèìî ñóïðîòèâíå, òîáòî ïðè x > x1<br />
ðÿä<br />
(12.5.1) çá³ãàºòüñÿ. Òîä³ çã³äíî ç äîâåäåíîþ ïåðøîþ ÷àñòèíîþ<br />
òåîðåìè ðÿä (12.5.1) ïîâèíåí çá³ãàòèñÿ ³ â òî÷ö³ x 1 ,<br />
òîìó ùî x1<br />
< x . Àëå öå ñóïåðå÷èòü óìîâ³, îñê³ëüêè â òî÷ö³<br />
x 1 ðÿä ðîçá³ãàºòüñÿ.<br />
Òåîðåìà Àáåëÿ ñòâåðäæóº, ùî ÿêùî x 0 — òî÷êà çá³æíîñò³<br />
ñòåïåíåâîãî ðÿäó, òî â óñ³õ òî÷êàõ, ðîçòàøîâàíèõ íà ³íòåðâàë³<br />
(–|x 0 |, |x 0 |), öåé ðÿä çá³ãàºòüñÿ àáñîëþòíî, à ÿêùî x 1 —<br />
òî÷êà ðîçá³æíîñò³ ñòåïåíåâîãî ðÿäó, òî â óñ³õ òî÷êàõ, ðîçòàøîâàíèõ<br />
ïîçà ³íòåðâàëó (–|x 1 |, |x 1 |), ðÿä ðîçá³ãàºòüñÿ.<br />
Ç òåîðåìè Àáåëÿ âèïëèâàº, ùî ÿêùî ðÿä (12.5.1) çá³ãà-<br />
ºòüñÿ íå ò³ëüêè ïðè x = 0, òî ³ñíóº ÷èñëî R > 0 òàêå, ùî ðÿä<br />
àáñîëþòíî çá³ãàºòüñÿ ïðè |x| R.<br />
²íòåðâàë (–R, R) íàçèâàºòüñÿ ³íòåðâàëîì çá³æíîñò³ ñòåïåíåâîãî<br />
ðÿäó. ×èñëî R íàçèâàºòüñÿ ðàä³óñîì çá³æíîñò³<br />
n<br />
n<br />
464 465
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