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Консультация № 199547
07.11.2020, 12:48
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе:
Найти область сходимости ряда
∑_(n=1)^∞[(x-1)^n/(2n^2 )]
Найти область сходимости ряда
∑_(n=1)^∞[(x-1)^n/(2n^2 )]
Прикрепленные файлы:
Обсуждение
12.11.2020, 05:54
общий
Адресаты:
Мне удалось найти замечательную статью по Вашей теме "Функциональные ряды. Степенные ряды. Область сходимости ряда" Ссылка
Однако, до закрытия Вашей консультации осталось менее 7 часов. За это время я не успеваю изучить статью и оформить полноценный Ответ.
Поэтому, если Вы ещё нуждаетесь в помощи, то просите модераторов продлить консультацию на пару суток. Либо дождитесь её авто-закрытия и создайте новую консультацию.
Однако, до закрытия Вашей консультации осталось менее 7 часов. За это время я не успеваю изучить статью и оформить полноценный Ответ.
Поэтому, если Вы ещё нуждаетесь в помощи, то просите модераторов продлить консультацию на пару суток. Либо дождитесь её авто-закрытия и создайте новую консультацию.
13.11.2020, 06:42
общий
это ответ
Здравствуйте, Barsik22!
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|<1, то есть ряд сходится при 0<x<2. Исследуем сходимость ряда на границе. При x = 2 имеем ряд
который, очевидно, сходится, так как степенной ряд вида
сходится при всех a > 1. При x = 0 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный ряд сходится на интервале [0, 2].
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|<1, то есть ряд сходится при 0<x<2. Исследуем сходимость ряда на границе. При x = 2 имеем ряд
который, очевидно, сходится, так как степенной ряд вида
сходится при всех a > 1. При x = 0 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный ряд сходится на интервале [0, 2].
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