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Консультация № 190522
03.02.2017, 20:07
0.00 руб.
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Найти область сходимости ряда
Найти область сходимости ряда
Прикрепленные файлы:
Обсуждение
04.02.2017, 14:29
общий
это ответ
Здравствуйте, verunchik_22!
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, расходится (т.н. гармонический ряд). При x = 0 имеем знакочередующийся ряд
который сходится по теореме Лейбница (знакочередующийся ряд сходится, если последовательность модулей его членов монотонно убывает и стремится к нулю). Следовательно, область сходимости исходного ряда - [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, расходится (т.н. гармонический ряд). При x = 0 имеем знакочередующийся ряд
который сходится по теореме Лейбница (знакочередующийся ряд сходится, если последовательность модулей его членов монотонно убывает и стремится к нулю). Следовательно, область сходимости исходного ряда - [0, 2).
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