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Консультация № 102764
20.09.2007, 14:11
0.00 руб.
0
1
1
3. При заданных значениях a и b написать первые три члена степенного ряда
∞
∑ a^n x^n /b^n * 3√n+1,
n=1
найти интервал сходимости ряда и исследовать его сходимости на концах интервала. a=6 и b=4
Приложение:
3√n+1 это корень третьей степени из n+1Помогите пожалуйста в течении полу часа экзамен через час !!!!!!!!!!!!!!!!!!
∞
∑ a^n x^n /b^n * 3√n+1,
n=1
найти интервал сходимости ряда и исследовать его сходимости на концах интервала. a=6 и b=4
Приложение:
3√n+1 это корень третьей степени из n+1Помогите пожалуйста в течении полу часа экзамен через час !!!!!!!!!!!!!!!!!!
Обсуждение
Неизвестный
20.09.2007, 14:33
общий
это ответ
Здравствуйте, Ерёменко Сергей Владимирович !
Общий член будет выглядеть так: un = (1.5x)^n/root[3](n+1).
Воспользуемся признаком Даламбера:
limit{n->беск}((1.5x)^(n+1)/root[3](n+2))/((1.5x)^n/root[3](n+1)) = limit{n->беск}(1.5x*root[3]((n+1)/(n+2))) = 1.5x
|1.5x| < 1 => |x| < 2/3
Значит, при -2/3 < x < 2/3 ряд сходится,
при x > 2/3 и x < -2/3 расходится.
Проверим при x=2/3:
un = 1/root[3](n+1) > 1/(n+1) - ряд расходится.
При x=-2/3:
un = (-1)^n/root[3](n+1) - знакочередующийся ряд, для его сходимости достаточно, чтобы
limit{n->беск}|un| = 0. Это выполняется:
limit{n->беск}(1/root[3](n+1)) = 1/беск = 0.
Значит, при x=-2/3 ряд сходится.
Общий член будет выглядеть так: un = (1.5x)^n/root[3](n+1).
Воспользуемся признаком Даламбера:
limit{n->беск}((1.5x)^(n+1)/root[3](n+2))/((1.5x)^n/root[3](n+1)) = limit{n->беск}(1.5x*root[3]((n+1)/(n+2))) = 1.5x
|1.5x| < 1 => |x| < 2/3
Значит, при -2/3 < x < 2/3 ряд сходится,
при x > 2/3 и x < -2/3 расходится.
Проверим при x=2/3:
un = 1/root[3](n+1) > 1/(n+1) - ряд расходится.
При x=-2/3:
un = (-1)^n/root[3](n+1) - знакочередующийся ряд, для его сходимости достаточно, чтобы
limit{n->беск}|un| = 0. Это выполняется:
limit{n->беск}(1/root[3](n+1)) = 1/беск = 0.
Значит, при x=-2/3 ряд сходится.
Форма ответа
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