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Задать вопрос экспертам

Консультация № 170501

17.07.2009, 09:35

0.00 руб.

0 3 1

Образование

Исследовать на сходимость знакопеременный ряд:
[$8721$](n=1...[$8734$])(-1)ⁿ⁺¹/(5n+3)

Обсуждение

Неизвестный

17.07.2009, 11:58

общий

это ответ

Здравствуйте, Alik4546.

Рассмотрим ряд ∑(n=1...∞)(-1)ⁿ⁺¹/(5n+3)=1/5*∑(n=1...∞)(-1)ⁿ⁺¹/(n+3/5).
Как известно знакочередующийся ряд ∑(n=1...∞)(-1)ⁿ⁺¹ 1/n сходится.
Скорость роста знаменателя определяется числом n.
Числа n и n+3/5 примерно равны при n[$8594$][$8734$], поэтому ряд ∑(n=1...∞)(-1)ⁿ⁺¹/(n+3/5) также будет сходиться.
Следовательно ряд ∑(n=1...∞)(-1)ⁿ⁺¹/(5n+3) сходится.

5

Неизвестный

17.07.2009, 12:30

общий

And0809:
And0809 позвольте уточнить ваш ответ
Ряд [$8721$]{n=1 ... [$8734$]}(-1)ⁿ⁺¹/(5n+3) сходится согласно признаку Лейбница, так как этот ряд является знакочередующимся и каждый последующий член ряда по модулю меньше предыдущего:
|a_n+1| - |a_n| = |(-1)ⁿ⁺¹⁺¹/(5(n+1)+3)| - |(-1)ⁿ⁺¹/(5n+3)| = (1/(5n+8)) - (1/(5n+3)) = -5/((5n+3)(5n+8)) < 0

Но расходится ряд, составленный из модулей исходного ряда, то есть ряд:
[$8721$]{n=1 ... [$8734$]} |(-1)ⁿ⁺¹/(5n+3)| = [$8721$]{n=1 ... [$8734$]}1/(5n+3)
Его достаточно сравнить с гармоническим рядом [$8721$]{n=1 ... [$8734$]}1/n

Значит исходный ряд сходится условно

Неизвестный

17.07.2009, 12:40

общий

Kom906:
Благодарю за уточнение!

Извиняюсь, не уточнил какая сходимость.

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