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Консультация № 198846
04.06.2020, 19:14
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Как найти область сходимости ряда
∑_(n=1)^∞▒(3ⁿ xⁿ)/∛n.
Как найти область сходимости ряда
∑_(n=1)^∞▒(3ⁿ xⁿ)/∛n.
Прикрепленные файлы:
Обсуждение
18.07.2022, 16:09
общий
это ответ
Здравствуйте, master87!
Используя признак Даламбера, получим
Определим, при каких значениях
этот предел меньше единицы: 
отсюда 
В этом промежутке заданный ряд сходится абсолютно.
При
получим числовой ряд с положительными членами 
который расходится как ряд Дирихле с показателем степени 
При
получим знакочередующийся числовой ряд 
который удовлетворяет признаку Лейбница и сходится условно, поскольку рассмотренный выше ряд, составленный из модулей его членов, расходится.
Следовательно, областью сходимости заданного ряда является промежуток
Используя признак Даламбера, получим
Определим, при каких значениях
этот предел меньше единицы: отсюда В этом промежутке заданный ряд сходится абсолютно.При
получим числовой ряд с положительными членами который расходится как ряд Дирихле с показателем степени При
получим знакочередующийся числовой ряд который удовлетворяет признаку Лейбница и сходится условно, поскольку рассмотренный выше ряд, составленный из модулей его членов, расходится.Следовательно, областью сходимости заданного ряда является промежуток
Об авторе:
Facta loquuntur.
Facta loquuntur.
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