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Консультация № 174617
28.11.2009, 09:18
35.00 руб.
0
1
1
Добрый день.
Найти область сходимости степенного ряда
[$8721$][$8734$]n=1 (x-2)n/2n
Заранее спасибо.
Найти область сходимости степенного ряда
[$8721$][$8734$]n=1 (x-2)n/2n
Заранее спасибо.
Обсуждение
Неизвестный
28.11.2009, 14:01
общий
29.11.2009, 01:31
это ответ
Здравствуйте, Александров Анатолий Олегович.
Воспользуемся признаком Даламбера для нахождения интервала сходимости данного ряда:
R = limn ->∞|2n/ (2(n+1))| = 1.
Следовательно, данный ряд абсолютно сходится на промежутке
|x-2|<1 или 1 < x < 3.
При x = 1 общий член знакопеременного ряда (-1)n/(2n+2) стремится к 0. Следовательно по признаку Лейбница ряд сходится условно.
При x = 3 исследуемый ряд является гармоническим и, следовательно, расходится.
Окончательный ответ: областью сходимости ряда является промежуток
1 ≤ x < 3.
Воспользуемся признаком Даламбера для нахождения интервала сходимости данного ряда:
R = limn ->∞|2n/ (2(n+1))| = 1.
Следовательно, данный ряд абсолютно сходится на промежутке
|x-2|<1 или 1 < x < 3.
При x = 1 общий член знакопеременного ряда (-1)n/(2n+2) стремится к 0. Следовательно по признаку Лейбница ряд сходится условно.
При x = 3 исследуемый ряд является гармоническим и, следовательно, расходится.
Окончательный ответ: областью сходимости ряда является промежуток
1 ≤ x < 3.
Форма ответа
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