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Консультация № 184946
24.12.2011, 15:55
60.71 руб.
0
1
1
Здравствуйте! Прошу помощи в следующем вопросе:
Исследовать сходимость данных числовых рядов
Приложение:
Исследовать сходимость данных числовых рядов
Приложение:
Обсуждение
24.12.2011, 16:37
общий
это ответ
Здравствуйте, Artek9300!
1. Воспользуемся признаком Лейбница достаточного условия сходимости знакочередующегося ряда: если последовательность a[sub]n[/sub] является монотонной и невозрастающей, причём a[sub]n[/sub] [$8594$] 0 при n [$8594$] 0, то ряд [$8721$](-1)[sup]n[/sup]a[sub]n[/sub] сходится. В данном случае для последовательности
в силу монотонного возрастания функций ln x и x[sup]2[/sup] при всех положительных x имеем a[sub]n+1[/sub] < a[sub]n[/sub] (монотонно убывающая последовательность), причём
Следовательно, исходный ряд
сходится (по признаку Лейбница). Поскольку ln[sup]2[/sup]n < n, [$8704$]n>1, то
и последний ряд расходится (гармонический ряд), то есть исходный ряд сходится условно, но не абсолютно.
2. Для данного ряда имеем
и так как последний ряд расходится (как ряд вида [$8721$]1/n[sup]a[/sup] при a=1/2<1), то исходный ряд также расходится.
1. Воспользуемся признаком Лейбница достаточного условия сходимости знакочередующегося ряда: если последовательность a[sub]n[/sub] является монотонной и невозрастающей, причём a[sub]n[/sub] [$8594$] 0 при n [$8594$] 0, то ряд [$8721$](-1)[sup]n[/sup]a[sub]n[/sub] сходится. В данном случае для последовательности
в силу монотонного возрастания функций ln x и x[sup]2[/sup] при всех положительных x имеем a[sub]n+1[/sub] < a[sub]n[/sub] (монотонно убывающая последовательность), причём
Следовательно, исходный ряд
сходится (по признаку Лейбница). Поскольку ln[sup]2[/sup]n < n, [$8704$]n>1, то
и последний ряд расходится (гармонический ряд), то есть исходный ряд сходится условно, но не абсолютно.
2. Для данного ряда имеем
и так как последний ряд расходится (как ряд вида [$8721$]1/n[sup]a[/sup] при a=1/2<1), то исходный ряд также расходится.
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