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Консультация № 203126
04.09.2022, 11:14
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:Найти интервал сходимости степеного ряда и исследовать сходимость на концах интервала сходимости:
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Обсуждение
04.09.2022, 12:41
общий
это ответ
Здравствуйте, mavrin-mpo!
Требуется определить область сходимости степенного ряда
Определим радиус сходимости заданного ряда, используя радикальный признак Коши:
Значит, заданный ряд абсолютно сходится при
Это и есть интервал сходимости ряда.
При
имеем ряд
который расходится, потому что для него не выполняется необходимое условие сходимости:
По той же причине расходится знакочередующийся ряд
который получается из заданного ряда при 
Следовательно, областью сходимости заданного ряда является промежуток То есть область сходимости ряда совпадает с интервалом сходимости.
Требуется определить область сходимости степенного ряда
Определим радиус сходимости заданного ряда, используя радикальный признак Коши:
Значит, заданный ряд абсолютно сходится при
Это и есть интервал сходимости ряда.При
имеем рядПо той же причине расходится знакочередующийся ряд
который получается из заданного ряда при Следовательно, областью сходимости заданного ряда является промежуток
То есть область сходимости ряда совпадает с интервалом сходимости.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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