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Консультация № 181019
29.11.2010, 18:10
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Здравствуйте, уважаемые эксперты! Прошу Вас ответить на следующий вопрос:
дан ряд [$8721$](n=1,[$8734$])1/ln(n+2)
Исследовать его на сходимость
дан ряд [$8721$](n=1,[$8734$])1/ln(n+2)
Исследовать его на сходимость
Обсуждение
29.11.2010, 19:15
общий
это ответ
Здравствуйте, Litta!
Воспользуемся теоремой сравнения: если для рядов с неотрицательными общими членами an и bn выполнено условие
an=o( bn)
и ряд с общим членом an расходится, то ряд с общим членом bn также расходится.
Так как lim(ln(n+2)/n)=0, то 1/n=o(1/ln(n+2)). Так как ряд с общим членом an=1/n расходится (гармонический ряд), то отюсда следует, что расходится и ряд с общим членом 1/ln(n+2) (теорема сравнения).
Воспользуемся теоремой сравнения: если для рядов с неотрицательными общими членами an и bn выполнено условие
an=o( bn)
и ряд с общим членом an расходится, то ряд с общим членом bn также расходится.
Так как lim(ln(n+2)/n)=0, то 1/n=o(1/ln(n+2)). Так как ряд с общим членом an=1/n расходится (гармонический ряд), то отюсда следует, что расходится и ряд с общим членом 1/ln(n+2) (теорема сравнения).
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