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Ïðèêëàä 5.1.12. Êîðèñòóþ÷èñü îçíà÷åííÿì 5.1.6, äîâåñòè,<br />
ùî ïîñë³äîâí³ñòü {q n } çà óìîâè, ùî q < 1, º íåñê³í÷åííî<br />
ìàëîþ.<br />
Ðîçâ’ÿçàííÿ. Çíîâó æ òàêè â³çüìåìî áóäü-ÿêå äîäàòíå<br />
÷èñëî ε. Êîðèñòóþ÷èñü çíàêîì ⇔, îòðèìàºìî òàê³ íåð³âíîñò³<br />
n<br />
n<br />
lg ε<br />
q N áóäå âèêîíóâàòèñÿ íåð³âí³ñòü n > , ùî ïðèâîäèòü<br />
çàâäÿêè âëàñòèâîñòÿì çíàêó ⇔ äî âèõ³äíî¿ íåð³âíî-<br />
lg q<br />
n<br />
ñò³ q N âèêîíóºòüñÿ íåð³âí³ñòü<br />
n<br />
q 0 ³ ïîêëàäåìî A = . Çã³äíî ε<br />
1<br />
ç îçíà÷åííÿì 5.1.5, äëÿ öüîãî À ³ñíóº íîìåð N òàêèé, ùî<br />
ïðè n>N âèêîíóºòüñÿ íåð³âí³ñòü x > A . Òîä³<br />
1 1 1<br />
α<br />
n<br />
= = < =ε,<br />
x x A<br />
n<br />
n<br />
òîáòî α<br />
n<br />
N.<br />
À öå îçíà÷àº, ùî<br />
⎧⎪<br />
1 ⎫<br />
ïîñë³äîâí³ñòü ⎨ ⎬<br />
⎪⎩ x<br />
— íåñê³í÷åííî ìàëà.<br />
n ⎭<br />
Àíàëîã³÷íî äîâîäèòüñÿ äðóãà ÷àñòèíà òåîðåìè.<br />
Ïðèêëàä 5.1.13. Äîâåñòè, ùî ïîñë³äîâí³ñòü {q n } º íåñê³í-<br />
÷åííî âåëèêîþ, ÿêùî |q|>1.<br />
1<br />
Äîâåäåííÿ. Ïîçíà÷èìî q<br />
1 = , òîä³ ÿêùî |q| >1,<br />
q<br />
1<br />
òî q1<br />
1.<br />
q = < Çã³äíî ç ïðèêëàäîì 5.1.12 ïîñë³äîâí³ñòü q }<br />
⎧⎪<br />
1 ⎫<br />
º íåñê³í÷åííî ìàëîþ, ³ ÿê íàñë³äîê ïîñë³äîâí³ñòü ⎨ ⎬<br />
⎪⎩ q<br />
n òåæ<br />
⎭<br />
áóäå íåñê³í÷åííî ìàëîþ. Çà äîâåäåíîþ òåîðåìîþ 5.1.1 ïîñë³äîâí³ñòü<br />
{q n } º íåñê³í÷åííî âåëèêîþ.<br />
n<br />
n<br />
{ 1<br />
138 139
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