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Консультация № 198400
26.04.2020, 20:57
0.00 руб.
26.04.2020, 22:21
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Найти область сходимости ряда
Найти область сходимости ряда
Прикрепленные файлы:
Обсуждение
27.04.2020, 12:34
общий
Обратите внимание на этот вопрос. Спасибо!
Об авторе:
Мне безразлично, что Вы думаете о обо мне, но я рад за Вас - Вы начали думать.
Мне безразлично, что Вы думаете о обо мне, но я рад за Вас - Вы начали думать.
01.05.2020, 14:45
общий
это ответ
Здравствуйте, Dina!
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
для которого
то есть не выполняется необходимое условие сходимости ряда lim a[sub]n[/sub] = 0. При x = -1 имеем знакочередующийся ряд
для которого необходимое условие сходимости также не выполняется. Следовательно, область сходимости исходного ряда - (-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
для которого
то есть не выполняется необходимое условие сходимости ряда lim a[sub]n[/sub] = 0. При x = -1 имеем знакочередующийся ряд
для которого необходимое условие сходимости также не выполняется. Следовательно, область сходимости исходного ряда - (-1, 1).
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