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Консультация № 197421
14.12.2019, 18:19
0.00 руб.
14.12.2019, 21:27
0
1
1
Здравствуйте! У меня возникли сложности с таким вопросом:
Обсуждение
20.12.2019, 04:16
общий
это ответ
Здравствуйте, Алиса!
Воспользуемся признаком Даламбера: если существует предел
то ряд [$8721$]a[sub]n[/sub] сходится при [$961$]<1 и расходится при [$961$]>1 (при [$961$]=1 сходимость ряда нужно исследовать особо). В данном случае
Из [$961$] = |x| <1 следует, что ряд сходится при -1 < x < 1 и расходится при x < -1 и при x > 1. При x = 1 получаем сходящийся ряд
(степенной ряд [$8721$]1/n[sup]a[/sup] сходится при всех a > 1). При x = -1 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный функциональный ряд сходится на отрезке [-1, 1].
Воспользуемся признаком Даламбера: если существует предел
то ряд [$8721$]a[sub]n[/sub] сходится при [$961$]<1 и расходится при [$961$]>1 (при [$961$]=1 сходимость ряда нужно исследовать особо). В данном случае
Из [$961$] = |x| <1 следует, что ряд сходится при -1 < x < 1 и расходится при x < -1 и при x > 1. При x = 1 получаем сходящийся ряд
(степенной ряд [$8721$]1/n[sup]a[/sup] сходится при всех a > 1). При x = -1 имеем знакочередующийся ряд
который является сходящимся по признаку Лейбница (последовательность его членов монотонно убывает и стремится к нулю). Следовательно, исходный функциональный ряд сходится на отрезке [-1, 1].
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