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Задать вопрос экспертам

Консультация № 183324

24.05.2011, 20:57

54.93 руб.

0 1 1

Образование

Уважаемые эксперты! Пожалуйста, ответьте на вопрос:Дан степенной ряд
∑от (n-1) до ∞ a^n*x^n/(b^n*√(n+1)).При заданных значениях a=7,b=6 написать три первых члена ряда и исследовать его сходимость на концах интервала.

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

24.05.2011, 21:36

общий

это ответ

Здравствуйте, Посетитель - 375555!
Для данных значениях параметра имеем ряд
[$8721$]_n=1^[$8734$](7ⁿ/(6ⁿ[$8730$](n+1)))xⁿ
Это степенной ряд с коэффициентами
c_n=7ⁿ/(6ⁿ[$8730$](n+1))

Три первых члена ряда:
a₁=7x/(6[$8730$]2)
a₂=49x²/(36[$8730$]3)
a₃=343x³/432

Радиус сходимости:
R=lim|c_n/c_n+1|=lim6[$8730$](n+2)/7[$8730$](n+1)=6/7

Сходимость в граничных точках:
1) x=-6/7, имеем ряд
[$8721$](-1)ⁿ/[$8730$](n+1)
Это знакочередующийся ряд с модулем общего члена 1/[$8730$](n+1) монотонно убывающим к нулю. По признаку Лейбница этот ряд сходится.
2) x=6/7, имеем ряд
[$8721$]1/[$8730$](n+1)
Это ряд степенного типа, с общим членом эквивалентным 1/[$8730$]n=1/n^1/2
Из общей теории известно, что ряд с общим членом 1/n^[$945$] сходится при [$945$]>1 и расходится при [$945$][$8804$]1. У нас [$945$]=1/2, следовательно, ряд расходится.

5

большое спасибо!

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