ЛЕКЦІЇ ² ВПРАВИ

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