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Консультация № 201145
10.06.2021, 22:57
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Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
исследовать на сходимость ряды:
исследовать на сходимость ряды:
Прикрепленные файлы:
Обсуждение
11.06.2021, 08:19
общий
это ответ
Здравствуйте, ushatalal!
Пусть требуется исследовать на сходимость ряд
Это знакочередующийся ряд, и для того, чтобы он сходился хотя бы условно, необходимо, чтобы общий член этого ряда стремился к нулю (одно из условий признака Лейбница сходимости знакочередующихся рядов), то есть чтобы выполнялось условие 
при 
Для проверки выполнения этого условия рассмотрим ряд 
каждый член которого по абсолютной величине меньше соответствующего члена заданного ряда (ведь 
для всех номеров 
). Поскольку при 
имеем 
постольку ряд 
расходится ввиду невыполнения необходимого условия сходимости. Значит, это условие не выполняется и для заданного ряда. Поэтому заданный ряд расходится.
Пусть требуется исследовать на сходимость ряд
Это знакочередующийся ряд, и для того, чтобы он сходился хотя бы условно, необходимо, чтобы общий член этого ряда стремился к нулю (одно из условий признака Лейбница сходимости знакочередующихся рядов), то есть чтобы выполнялось условие при Для проверки выполнения этого условия рассмотрим ряд каждый член которого по абсолютной величине меньше соответствующего члена заданного ряда (ведь для всех номеров ). Поскольку при имеем постольку ряд расходится ввиду невыполнения необходимого условия сходимости. Значит, это условие не выполняется и для заданного ряда. Поэтому заданный ряд расходится.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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