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Задать вопрос экспертам

Консультация № 177773

09.04.2010, 12:38

40.27 руб.

0 1 1

Образование

Здравствуйте! Помогите, пожалуйста, исследовать на сходимость и такой ряд (по признаку Коши и, если возможно, поподробнее):

Спасибо!

Обсуждение

Неизвестный

09.04.2010, 14:32

общий

это ответ

Здравствуйте, MrSpencer.

Признак Коши сходимости рядов:

Если с некоторого номера n выполняется (a_n)^1/n< d и 0<d<1, то ряд [$8721$]a_n сходится.
Т.е. достаточно показать , что lim_n->[$8734$](a_n)^1/n<1

[$8721$](a*cos²(Pi*n/3))/2ⁿ=a*[$8721$](cos²(Pi*n/3))/2ⁿ
lim_n->[$8734$]((cos²(Pi*n/3))/2ⁿ)^1/n=(1/2)* lim_n->[$8734$](cos²(Pi*n/3))^1/n
Т.к. cos²(Pi*n/3)[$8804$]1 для любых n, то
(1/2)* lim_n->[$8734$](cos²(Pi*n/3))^1/n[$8804$](1/2)*1
Получится, что для любого числа d, 1/2<d<1 будет выполняться (a_n)^1/n< d
следовательно ряд [$8721$](cos²(Pi*n/3))/2ⁿ сходится и ряд [$8721$](a*cos²(Pi*n/3))/2ⁿ тоже сходится

5

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