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Íåð³âí³ñòü (12.2.13) ãîâîðèòü ïðî òå, ùî ïîñë³äîâí³ñòü a n<br />
çðîñòຠ³ òèì ñàìèì íå âèêîíóºòüñÿ íåîáõ³äíà óìîâà çá³æíîñò³<br />
ðÿäó. Òàêèì ÷èíîì, ðÿä (12.2.8) ó öüîìó âèïàäêó<br />
ðîçá³ãàºòüñÿ.<br />
3°. ßêùî ρ = 1, òî íà ïðèêëàäàõ ìîæíà ïîêàçàòè, ùî ðÿä<br />
â îäíèõ âèïàäêàõ çá³ãàºòüñÿ, à â ³íøèõ âèïàäêàõ ðîçá³ãà-<br />
ºòüñÿ. ßñíî, ùî â òàêèõ âèïàäêàõ ìè ïîâèíí³ çàñòîñóâàòè<br />
³íøó îçíàêó çá³æíîñò³.<br />
Òåîðåìó äîâåäåíî.<br />
Ïðèêëàä 12.2.1. Äîñë³äèòè çà îçíàêîþ Äàëàìáåðà çá³æí³ñòü<br />
ðÿäó ∑<br />
∞<br />
5<br />
n<br />
n<br />
n=<br />
1 2 .<br />
Ðîçâ’ÿçàííÿ. Çíàþ÷è n-é ÷ëåí ðÿäó, çíàõîäèìî íàñòóïíèé<br />
çà íèì (n + 1)-é ÷ëåí, çàì³íþþ÷è â âèðàç³ n-ãî<br />
÷ëåíà n íà n + 1. Ïîò³ì øóêàºìî ãðàíèöþ â³äíîøåííÿ íàñòóïíîãî<br />
÷ëåíà a n+1 äî ïîïåðåäíüîãî a n ïðè íåîáìåæåíîìó<br />
çðîñòàíí³ n:<br />
( n + 1)<br />
5<br />
n<br />
an<br />
= , a<br />
n n +<br />
=<br />
n<br />
2 2 +<br />
1 1<br />
an+<br />
1<br />
1⎛n<br />
+ 1⎞ 1 ⎛ 1⎞<br />
1<br />
ρ= lim = lim ⎜ ⎟ = lim ⎜1+ ⎟ =<br />
n→+∞ a n→+∞ n<br />
n<br />
2 n →+∞<br />
⎝ ⎠ 2 ⎝ n⎠<br />
2 .<br />
Òóò ρ < 1. Òîìó çã³äíî ç îçíàêîþ Äàëàìáåðà äàíèé ðÿä<br />
çá³ãàºòüñÿ.<br />
Íàâåäåìî, áåç äîâåäåííÿ, ³íøó äîñòàòíþ îçíàêó çá³æíîñò³<br />
ðÿäó ç äîäàòíèìè ÷ëåíàìè — ³íòåãðàëüíó îçíàêó Êîø³.<br />
Òåîðåìà 12.2.2. Ðÿä ç äîäàòíèìè ñïàäíèìè ÷ëåíàìè<br />
a n = f(n) çá³ãàºòüñÿ àáî ðîçá³ãàºòüñÿ, â çàëåæíîñò³ â³ä òîãî,<br />
5<br />
5 5<br />
çá³ãàºòüñÿ ÷è ðîçá³ãàºòüñÿ íåâëàñíèé ³íòåãðàë<br />
f(x) — íåïåðåðâíà ñïàäíà ôóíêö³ÿ.<br />
Ïðèêëàä 12.2.2. Äîñë³äèòè çá³æí³ñòü ðÿäó<br />
+ 1 + 1 + ... + 1 ∞<br />
...<br />
α α α α<br />
( )<br />
n<br />
+ = 1<br />
1 ∑<br />
n=<br />
n<br />
α > 0 .<br />
2 3<br />
1<br />
;<br />
∞<br />
∫ fxdx ( ) , äå<br />
1<br />
Ðîçâ’ÿçàííÿ. Íåâàæêî ïåðåêîíàòèñÿ (äèâ. ïðèêë.<br />
∞<br />
dx<br />
9.6.6), ùî íåâëàñíèé ³íòåãðàë ∫ α ïðè α > 1 çá³ãàºòüñÿ, à<br />
1 x<br />
ïðè α≤1 ðîçá³ãàºòüñÿ. Îòæå, äàíèé ðÿä çá³ãàºòüñÿ ïðè α >1<br />
òà ðîçá³ãàºòüñÿ ïðè α≤1.<br />
Çàóâàæåííÿ. Ïðè α=1 ðÿä íàáóâຠâèãëÿäó:<br />
1 1 1 ∞ 1<br />
1+ + + ... + + ... = ∑ . (12.2.14)<br />
2 3 n n=<br />
1 n<br />
Ðÿä (12.2.14) íàçèâàºòüñÿ ãàðìîí³÷íèì. Îñê³ëüêè, α=1,<br />
òî ãàðìîí³÷íèé ðÿä ðîçá³ãàºòüñÿ. Äî ðå÷³, çà îçíàêîþ Äàëàìáåðà<br />
öå íåìîæëèâî âñòàíîâèòè. ×èòà÷åâ³ ðåêîìåíäóºìî<br />
öåé ôàêò îá´ðóíòóâàòè.<br />
ÂÏÐÀÂÈ<br />
Äîñë³äèòè çá³æí³ñòü ðÿä³â:<br />
∞ 1<br />
∞ 1<br />
12.1. ∑<br />
3 ; 12.2. ∑<br />
n=<br />
2 nln<br />
n<br />
n = 0 4n<br />
+ 1 ; 12.3. ∞ n<br />
∑<br />
4<br />
n=<br />
3 n − 9 ;<br />
∞ 1 ∞ 1<br />
12.4. ∑<br />
n = n 2 ; 12.5. ∑<br />
2<br />
1<br />
n = 2 n −1 ; 12.6. ∞ n!<br />
∑<br />
n<br />
n=<br />
1 5 ;<br />
∞ 1<br />
∞<br />
2n+<br />
1<br />
∞<br />
3n<br />
12.7. ∑ 3<br />
7<br />
n=<br />
1 ( n + 1)<br />
n<br />
; 12.8. ∑<br />
3n−1<br />
; 12.9. ∑<br />
n=<br />
0<br />
n=<br />
3<br />
2<br />
( 2n<br />
− 5 )!<br />
;<br />
∞<br />
3<br />
∞<br />
n<br />
1<br />
∞<br />
12.10. ∑<br />
2n<br />
−1<br />
∑<br />
n=<br />
1 ( 2n<br />
; 12.11. ∑<br />
)!<br />
n<br />
; 12.12. n=<br />
2 3<br />
n=<br />
1 2<br />
2n<br />
3<br />
∞<br />
n<br />
3<br />
∞<br />
12.13. ∑<br />
n<br />
n=<br />
0 2 ( 2n<br />
+ 1 )<br />
; 12.14. 4n<br />
− 3<br />
∑<br />
n=<br />
1 n<br />
n3 ; 12.15. ∞ 1<br />
∑<br />
n ;<br />
n=<br />
1 n<br />
1<br />
n=<br />
0<br />
( )<br />
− 2<br />
12.16.<br />
∞<br />
∞<br />
1<br />
∞<br />
∑ ; 12.17. ∑<br />
n=<br />
2 ln n<br />
( n + 13 ) n 1<br />
∑<br />
n = 0<br />
3<br />
n + 1 ;<br />
12.19.<br />
∞ 1<br />
∑<br />
n<br />
n=<br />
1 n5 ∞<br />
∞<br />
1<br />
ln ( n + 1)<br />
∑<br />
n<br />
n=<br />
0 2 + 1 ; 12.21. ∑<br />
n=<br />
1 3<br />
n 2 .<br />
Âêàç³âêà.  ïðèêëàäàõ 12.1 – 12.7 äîö³ëüíî âèêîðèñòîâóâàòè<br />
³íòåãðàëüíó îçíàêó çá³æíîñò³; 12.8 – 12.14 — îçíàêó<br />
Äàëàìáåðà; 12.15 – 12.21 — îçíàêó ïîð³âíÿííÿ.<br />
;<br />
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