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Ìîæíà äîâåñòè, ùî ÿêùî çá³ãàºòüñÿ ðÿä (12.4.2), òî é<br />
çá³ãàºòüñÿ ðÿä (12.4.1). Ó çâ’ÿçêó ç öèì ââîäèòüñÿ ïîíÿòòÿ<br />
àáñîëþòíî¿ çá³æíîñò³.<br />
Ðÿä (12.4.1) íàçèâàºòüñÿ àáñîëþòíî çá³æíèì, ÿêùî çá³ãà-<br />
ºòüñÿ ðÿä, ñêëàäåíèé ç àáñîëþòíèõ çíà÷åíü éîãî ÷ëåí³â,<br />
òîáòî ðÿä (12.4.2).<br />
Ðÿä (12.4.1) íàçèâàºòüñÿ óìîâíî çá³æíèì, ÿêùî â³í çá³ãà-<br />
ºòüñÿ, à ðÿä (12.4.2) ñêëàäåíèé ç àáñîëþòíèõ âåëè÷èí ÷ëåí³â<br />
ðÿäó ðîçá³ãàºòüñÿ.<br />
Ïðèêëàä 12.4.1. Ðÿä<br />
1 1 1 n+ 1 1 ∞ n+<br />
1 1<br />
1− + − + K+ ( − 1) + K= n<br />
∑ ( −1)<br />
n<br />
2 4 8 2 n=<br />
1 2<br />
º àáñîëþòíî çá³æíèì, òîìó ùî ðÿä, ñêëàäåíèé ç àáñîëþòíèõ<br />
âåëè÷èí,<br />
1 1 1 1 1<br />
∞<br />
1+ + + + K+ + K= n<br />
∑<br />
2 4 8 2<br />
n<br />
n=<br />
1 2<br />
òàêîæ çá³ãàºòüñÿ (îáèäâà ðÿäè — ãåîìåòðè÷í³ ïðîãðåñ³¿ ³ç<br />
çíàìåííèêàìè, â³äïîâ³äíî ð³âíèìè − 1 2 ³ 1 2 ).<br />
Ïðèêëàä 12.4.2. Ðÿä<br />
1 1 1 n+ 1 1 ∞ n+<br />
1 1<br />
1− + − + K+ ( − 1) + K= ∑( −1)<br />
2 3 4<br />
n n=<br />
1 n<br />
º óìîâíî çá³æíèì, òîìó ùî ñàì â³í çá³ãàºòüñÿ çà îçíàêîþ<br />
Ëåéáí³öà, à ðÿä ñêëàäåíèé ç àáñîëþòíèõ âåëè÷èí,<br />
1 1 1 1 ∞ 1<br />
1+ + + + K+ + K=∑<br />
2 3 4 n n=<br />
1 n<br />
ðîçá³ãàºòüñÿ (äèâ. ïðèêëàä 12.1.4).<br />
Çàóâàæåííÿ. Ïðè ïðàêòè÷íîìó âèêîðèñòàíí³ ðÿä³â<br />
(çá³æíèõ) çâè÷àéíî îáìåæóþòüñÿ ê³ëüêîìà ¿õ ïåðøèìè ÷ëåíàìè.<br />
Äîïóùåíà ïðè öüîìó ïîõèáêà (çàëèøîê ðÿäó) íàéá³ëüøå<br />
ïðîñòî îö³íþºòüñÿ äëÿ çíàêîïåðåì³æíèõ ðÿä³â: ïîõèáêà<br />
ïðè çàì³í³ ñóìè çá³æíîãî çíàêîïåðåì³æíîãî ðÿäó<br />
ñóìîþ n éîãî ïåðøèõ ÷ëåí³â ìåíøå àáñîëþòíîãî çíà÷åííÿ<br />
ïåðøîãî ç â³äêèíóòèõ ÷ëåí³â ðÿäó, òîáòî a n+1 . Äîâåäåííÿ<br />
öüîãî ôàêòó àíàëîã³÷íî ïðèéîìîâ³, âèêîðèñòàíîìó ïðè äîâåäåíí³<br />
îçíàêè Ëåéáí³öà.<br />
ÂÏÐÀÂÈ<br />
Äîñë³äèòè çá³æí³ñòü çíàêîïåðåì³æíèõ ðÿä³â. Âèçíà÷èòè,<br />
÷è º âîíè àáñîëþòíî çá³æíèìè, óìîâíî çá³æíèìè àáî ðîçá³æíèìè.<br />
( ) n −1<br />
n<br />
∞ −1<br />
∞ ( )<br />
12.22. ∑<br />
n=<br />
1 2n<br />
−1 ; 12.23.<br />
−1<br />
∑<br />
n=<br />
1 n( n+<br />
1 )<br />
; 12.24. ∞ ( −1) n<br />
∑<br />
3<br />
n=<br />
1 n + 1 ;<br />
( ) n −1<br />
∞ −1<br />
∞ ( −1) n<br />
12.25. ∑ ; 12.26. ∑<br />
n=<br />
1<br />
3<br />
n n<br />
n=<br />
0 2n<br />
+ 1 .<br />
12.5. ÏÎÍßÒÒß ÏÐÎ ÔÓÍÊÖ²ÎÍÀËÜͲ<br />
ÐßÄÈ<br />
Ðÿä<br />
( ) + ( ) + ( ) + + ( ) + = ∑ ( )<br />
∞<br />
f x f x f x f x f x<br />
1 2 3<br />
K<br />
n<br />
K<br />
n ,<br />
n=<br />
1<br />
÷ëåíè ÿêîãî º ôóíêö³ÿìè â³ä çì³ííî¿ x, º ôóíêö³îíàëüíèé<br />
ðÿä. ßêùî ôóíêö³¿ f n (x) ä³éñí³, òî ïðè ð³çíèõ çíà÷åííÿõ<br />
çì³ííî¿ x ³ç ôóíêö³îíàëüíîãî ðÿäó îäåðæóþòüñÿ ð³çí³ ÷èñëîâ³<br />
ðÿäè, ÿê³ ìîæóòü áóòè çá³æíèìè àáî ðîçá³æíèìè.<br />
12.5.1. Ñòåïåíåâ³ ðÿäè<br />
Ç óñ³õ ôóíêö³îíàëüíèõ ðÿä³â íàéïðîñò³øèìè ³ íàéá³ëüøå<br />
âæèâàíèìè º ñòåïåíåâ³ ðÿäè âèãëÿäó<br />
∞<br />
2<br />
n<br />
n<br />
a0 + a1x+ a2x + + anx + = ∑ anx<br />
n=<br />
0<br />
K K . (12.5.1)<br />
×èñëà a 0 , a 1 , a 2 , ..., a n , ... íàçèâàþòüñÿ êîåô³ö³ºíòàìè<br />
ñòåïåíåâîãî ðÿäó.<br />
Íàäàþ÷è x ð³çí³ ÷èñëîâ³ çíà÷åííÿ, áóäåìî îòðèìóâàòè<br />
ð³çí³ ÷èñëîâ³ ðÿäè, ÿê³ ìîæóòü âèÿâèòèñÿ çá³æíèìè àáî<br />
ðîçá³æíèìè. Ìíîæèíà òèõ çíà÷åíü x, ïðè ÿêèõ ðÿä (12.5.1)<br />
çá³ãàºòüñÿ, íàçèâàºòüñÿ îáëàñòþ éîãî çá³æíîñò³. Öÿ ìíîæè-<br />
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