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an = Sn −S n−1<br />
. Îñê³ëüêè S n → S ³ S n–1 → S ïðè n →∞ (ðÿä çá³ãàºòüñÿ),<br />
òî lim an = lim( Sn − Sn−1)<br />
= lim S0 − lim Sn−1<br />
= S − S = 0 .<br />
n→∞ n→∞ n→∞ n→∞<br />
Óìîâà lim a n<br />
= 0 º íåîáõ³äíîþ, àëå íå äîñòàòíüîþ óìîâîþ<br />
n→∞<br />
çá³æíîñò³ ðÿäó. Ïîêàæåìî öå íà ïðèêëàä³.<br />
lim a<br />
n→∞<br />
Ïðèêëàä 12.1.4. Ðîçãëÿíåìî ðÿä<br />
n<br />
1 1 1<br />
1+ + + + +<br />
2 3 K n<br />
K ,<br />
1<br />
= lim = 0 (âèêîíóºòüñÿ íåîáõ³äíà óìîâà). Ïðîòå ðÿä<br />
n→∞<br />
n<br />
ðîçá³ãàºòüñÿ.<br />
ijéñíî, îñê³ëüêè<br />
1 1 1 1 1 1 n<br />
Sn<br />
= 1+ + + + > + + + = = n<br />
2 3 K n n n K .<br />
n n<br />
òî lim S n<br />
=∞, òîáòî ðîçãëÿäóâàíèé ðÿä ðîçá³ãàºòüñÿ.<br />
n→∞<br />
12.2. ÄÎÑÒÀÒͲ ÎÇÍÀÊÈ ÇÁ²ÆÍÎÑÒ² ÐßIJÂ<br />
Ç ÄÎÄÀÒÍÈÌÈ ×ËÅÍÀÌÈ<br />
12.2.1. Äîïîì³æíå òâåðäæåííÿ<br />
Ïåðåä äîâåäåííÿì äîñòàòí³õ îçíàê çá³æíîñò³ ðÿä³â äîâåäåìî<br />
îäíó ëåìó.<br />
Ëåìà. ßêùî ó ðÿä³<br />
a 1 + a 2 + a 3 + ... + a n + ... (12.2.1)<br />
â³äêèíóòè ñê³í÷åííå ÷èñëî ïåðøèõ ïî÷àòêîâèõ ÷ëåí³â, íàïðèêëàä<br />
k ÷ëåí³â, òî îòðèìàºìî ðÿä<br />
a k+1 + a k+2 + a k+3 + ... + a k+n + ..., (12.2.2)<br />
ÿêèé çá³ãàºòüñÿ (àáî ðîçá³ãàºòüñÿ) îäíî÷àñíî ç äàíèì ðÿäîì<br />
(12.2.1).<br />
Äîâåäåííÿ. Ïîçíà÷èìî ñóìó â³äêèíåíèõ ÷ëåí³â ÷åðåç<br />
U:<br />
U = a 1 + a 2 + a 3 + ... + a k .<br />
Íåõàé S n — ñóìà ïåðøèõ n ÷ëåí³â ðÿäó (12.2.1), à σ n —<br />
ñóìà ïåðøèõ n ÷ëåí³â ðÿäó (12.2.2). Òîä³ ìàòèìåìî<br />
S n+k = U + σ n . (12.2.3)<br />
Çâ³äñè<br />
σ n = S n+k – U. (12.2.4)<br />
Íåõàé çá³ãàºòüñÿ ðÿä (12.2.1):<br />
lim S n<br />
= S . (12.2.5)<br />
n→∞<br />
Î÷åâèäíî, ùî ³ç ð³âíîñò³ (12.2.5) âèïëèâຠòàêà:<br />
lim<br />
=<br />
S n + k<br />
S.<br />
n→∞<br />
Ïåðåõîäÿ÷è äî ãðàíèöü (n →∞) ³ç ð³âíîñò³ (12.2.4) áóäåìî<br />
ìàòè<br />
σ = S – U,<br />
äå<br />
lim σ = σ<br />
n .<br />
n →∞<br />
Àíàëîã³÷íî äîâîäèòüñÿ, ùî ³ç çá³æíîñò³ ðÿäó (12.2.2) âèïëèâàº<br />
çá³æí³ñòü (12.2.1), ïðè÷îìó<br />
S = σ + U.<br />
Îòæå, ëåìó äîâåäåíî.<br />
12.2.2. Îçíàêà ïîð³âíÿííÿ ðÿä³â<br />
ßêùî ÷ëåíè ðÿäó<br />
a 1 + a 2 + a 3 + ... + a n + ... (12.2.6)<br />
íåâ³ä’ºìí³ ³ íå ïåðåâåðøóþòü â³äïîâ³äíèõ ÷ëåí³â çá³æíîãî<br />
ðÿäó<br />
b 1 + b 2 + b 3 + ... + b n + ..., (12.2.7)<br />
òî äàíèé ðÿä (12.2.6) òåæ çá³ãàºòüñÿ.<br />
Äîâåäåííÿ. Ïîçíà÷èìî<br />
S n = a 1 + a 2 + a 3 + ... + a n ,<br />
σ n =b 1 + b 2 + b 3 + ... + b n .<br />
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