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Консультация № 199639
18.11.2020, 09:11
0.00 руб.
1
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1
Здравствуйте! Прошу помощи в следующем вопросе:
Найти область сходимости ряда:
(x^n)/(n^2+n)
Найти область сходимости ряда:
(x^n)/(n^2+n)
Прикрепленные файлы:
Обсуждение
23.11.2020, 04:02
общий
это ответ
Здравствуйте, lyskov.kirill!
Воспользуемся признаком Даламбера: для степенного ряда
радиус сходимости определяется выражением
то есть ряд сходится при |x-x[sub]0[/sub]| < R, расходится при |x-x[sub]0[/sub]| > R, при |x-x[sub]0[/sub]| = R ряд может как сходиться, так и расходиться.
В данном случае
и
Отсюда |x|<1, то есть ряд сходится при -1<x<1. Исследуем сходимость ряда на границе. При x = 1 имеем ряд
который, очевидно, сходится, так как степенной ряд вида
сходится при всех a > 1. При x = -1 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный ряд сходится на интервале [-1, 1].
Воспользуемся признаком Даламбера: для степенного ряда
радиус сходимости определяется выражением
то есть ряд сходится при |x-x[sub]0[/sub]| < R, расходится при |x-x[sub]0[/sub]| > R, при |x-x[sub]0[/sub]| = R ряд может как сходиться, так и расходиться.
В данном случае
и
Отсюда |x|<1, то есть ряд сходится при -1<x<1. Исследуем сходимость ряда на границе. При x = 1 имеем ряд
который, очевидно, сходится, так как степенной ряд вида
сходится при всех a > 1. При x = -1 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный ряд сходится на интервале [-1, 1].
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