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S n<br />
1 1 1<br />
= + + K +<br />
1⋅2 2⋅ 3 n n + 1<br />
.<br />
( )<br />
Äîäàíêè ö³º¿ ñóìè ìîæóòü áóòè ïîäàí³ ó âèãëÿä³:<br />
Òîìó<br />
1 1 1 1 1 1 1 1<br />
= 1 − ; = − ; K ; = −<br />
1⋅2 2 2⋅ 3 2 3 n n + 1 n n + 1<br />
.<br />
( )<br />
⎛ 1⎞ ⎛1 1⎞ ⎛1 1⎞ ⎛1 1 ⎞ 1<br />
S n<br />
= ⎜1− 1<br />
2<br />
⎟ + ⎜ −<br />
2 3<br />
⎟ + ⎜ − + + − = −<br />
3 4<br />
⎟ ⎜<br />
n n 1<br />
⎟<br />
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ K ⎝ + ⎠ n + 1<br />
.<br />
Çâ³äñè âèïëèâàº, ùî ãðàíèöÿ ïîñë³äîâíîñò³ ÷àñòèííèõ<br />
ñóì äàíîãî ðÿäó äîð³âíþº îäèíèö³:<br />
⎛ 1 ⎞<br />
1<br />
lim Sn<br />
= lim 1 1 lim 1<br />
n→∞ n→∞ ⎜ −<br />
n 1<br />
⎟ = − = .<br />
n→∞<br />
⎝ + ⎠ n + 1<br />
Òàêèì ÷èíîì, ðÿä çá³ãàºòüñÿ ³ éîãî ñóìà äîð³âíþº îäèíèö³.<br />
Ïðèêëàä 12.1.2. Óñòàíîâèìî çá³æí³ñòü àáî ðîçá³æí³ñòü<br />
ðÿäó<br />
n<br />
∞<br />
( ) ∑ ( )<br />
− 1 n+<br />
1<br />
1− 1+ 1− 1+ K+ − 1 + K = −1<br />
.<br />
Ïîñë³äîâí³ñòü éîãî ÷àñòèííèõ ñóì ìຠâèãëÿä S 1 =1,<br />
S 2 = 0, S 3 = 1, S 4 = 0, ... . Öå îçíà÷àº, ùî ðÿä íå çá³ãàºòüñÿ<br />
í³ äî ÿêî¿ ãðàíèö³, òîìó äàíèé ðÿä ðîçá³ãàºòüñÿ.<br />
Ïðèêëàä 12.1.3. Ðîçãëÿíåìî ðÿä, ñêëàäåíèé ç ÷ëåí³â ãåîìåòðè÷íî¿<br />
ïðîãðåñ³¿:<br />
n=<br />
1<br />
2 n−1 ∞<br />
∑<br />
n−1<br />
n=<br />
1<br />
a + aq + aq + K+ aq + K = aq , a ≠ 0. (12.1.3)<br />
×àñòèííà ñóìà S n öüîãî ðÿäó ïðè q ≠ 1 ìຠâèãëÿä:<br />
n<br />
n<br />
2 n−1<br />
a − aq a aq<br />
Sn<br />
= a + aq + aq + K + aq = = − .<br />
1−q 1−q 1−q<br />
Çâ³äñè:<br />
n<br />
1) ÿêùî q < 1, òî lim lim a lim aq a<br />
Sn<br />
= − = , òîáòî ðÿä<br />
n→∞ n→∞1−q n→∞1−q 1−q<br />
çá³ãàºòüñÿ ³ éîãî ñóìà<br />
q = 1/5 ìàºìî:<br />
a<br />
S = 1 − q<br />
. Íàïðèêëàä, ïðè a =1,<br />
1 1 1 5<br />
S = + + + K+ + K = ;<br />
1 5 5<br />
2 5<br />
n−1<br />
4<br />
n<br />
2) ÿêùî q > 1, òî lim lim a−<br />
aq<br />
Sn<br />
= =∞, òîáòî ðÿä (12.1.3)<br />
n→∞<br />
n→∞<br />
1− q<br />
ðîçá³ãàºòüñÿ;<br />
3) ïðè q = 1 ðÿä (12.1.3) ïðèéìຠâèä: a + a + a +...+ a +...<br />
⎛<br />
⎞<br />
Ó öüîìó âèïàäêó lim Sn<br />
= lim a+ a+ + a = lim na =∞<br />
n→∞ n→∞⎜1 42443 K<br />
⎟<br />
, òîáòî<br />
n→∞<br />
⎝ n ðàç ⎠<br />
ðÿä ðîçá³ãàºòüñÿ;<br />
4) ïðè q = –1 ðÿä (12.1.3) ïðèéìຠâèãëÿä: a – a + a –<br />
a a( −1) n<br />
– a + ... + (–1) n–1 a + ... Äëÿ íüîãî Sn<br />
= − , òîáòî S n =0<br />
2 2<br />
ïðè n ïàðíîìó ³ S n = a ïðè n íåïàðíîìó. Îòæå, lim S n<br />
íå<br />
n→∞<br />
³ñíóº ³ ðÿä (12.1.3) ðîçá³ãàºòüñÿ. Òàêèì ÷èíîì, ðÿä (12.1.3)<br />
º çá³æíèì ïðè q < 1 ³ ðîçá³æíèì ïðè q ≥ 1 .<br />
12.1.2. Íåîáõ³äíà óìîâà çá³æíîñò³ ðÿäó<br />
Òåîðåìà 12.1.1 (ïðî íåîáõ³äíó óìîâó çá³æíîñò³ ðÿäó).<br />
ßêùî ðÿä<br />
∞<br />
∑ a<br />
n=<br />
1<br />
n<br />
çá³ãàºòüñÿ, òî ãðàíèöÿ çàãàëüíîãî ÷ëåíà<br />
ðÿäó äîð³âíþº íóëþ, òîáòî lim a n<br />
= 0 .<br />
n→∞<br />
Äîâåäåííÿ. Ðîçãëÿíåìî ÷àñòèíí³ ñóìè S n = a 1 + a 2 +<br />
+ a 3 + ... + a n ³ S n–1 = a 1 + a 2 + a 3 + ... + a n–1 . Î÷åâèäíî, ùî<br />
452 453
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