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Консультация № 160974
20.02.2009, 12:21
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Уважаемые эксперты!
Помогите, пожалуйста, исследовать на сходимость числовой ряд:
∑ (n+5)*sin(2/(3^n)) / (n!), где n от 1 до ∞.
Помогите, пожалуйста, исследовать на сходимость числовой ряд:
∑ (n+5)*sin(2/(3^n)) / (n!), где n от 1 до ∞.
Обсуждение
Неизвестный
26.02.2009, 02:28
общий
это ответ
Здравствуйте, Orxideja!
an = (n+5)*sin(2/3n) / n!.
Воспользуемся признаком Даламбера.
an+1 / an =
= (n+6)*sin(2/3n+1)/(n+1)! * n!/[(n+5)*sin(2/3n)] =
= (n+6)/(n+5) * n!/(n+1)! * sin(2/3n+1)/sin(2/3n) =
= (n+6)/[(n+5)(n+1)] * sin(2/3n+1)/sin(2/3n).
limn[$8594$][$8734$]an+1 / an =
= limn[$8594$][$8734$](n+6)/[(n+5)(n+1)] * sin(2/3n+1)/sin(2/3n) =
= limn[$8594$][$8734$](n+6)/[(n+5)(n+1)] * limn[$8594$][$8734$]sin(2/3n+1)/sin(2/3n) =
= 0 * limn[$8594$][$8734$]sin(2/3n+1)/(2/3n+1) * limn[$8594$][$8734$](2/3n)/sin(2/3n) * limn[$8594$][$8734$](2/3n+1)/(2/3n) =
= 0 * lim2/3^(n+1)[$8594$]0sin(2/3n+1)/(2/3n+1) * lim2/3^n[$8594$]0(2/3n)/sin(2/3n) * 1/3 =
= 0 * 1 * 1/1 * 1/3 =
= 0 < 1.
Ряд сходится.
an = (n+5)*sin(2/3n) / n!.
Воспользуемся признаком Даламбера.
an+1 / an =
= (n+6)*sin(2/3n+1)/(n+1)! * n!/[(n+5)*sin(2/3n)] =
= (n+6)/(n+5) * n!/(n+1)! * sin(2/3n+1)/sin(2/3n) =
= (n+6)/[(n+5)(n+1)] * sin(2/3n+1)/sin(2/3n).
limn[$8594$][$8734$]an+1 / an =
= limn[$8594$][$8734$](n+6)/[(n+5)(n+1)] * sin(2/3n+1)/sin(2/3n) =
= limn[$8594$][$8734$](n+6)/[(n+5)(n+1)] * limn[$8594$][$8734$]sin(2/3n+1)/sin(2/3n) =
= 0 * limn[$8594$][$8734$]sin(2/3n+1)/(2/3n+1) * limn[$8594$][$8734$](2/3n)/sin(2/3n) * limn[$8594$][$8734$](2/3n+1)/(2/3n) =
= 0 * lim2/3^(n+1)[$8594$]0sin(2/3n+1)/(2/3n+1) * lim2/3^n[$8594$]0(2/3n)/sin(2/3n) * 1/3 =
= 0 * 1 * 1/1 * 1/3 =
= 0 < 1.
Ряд сходится.
Форма ответа
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