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Консультация № 111914
01.12.2007, 21:17
0.00 руб.
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Доброго времени суток! Мне нужна ваша помощь в решении задачи
Исследовать на сходимость ряд
сумма от 2 до бесконечности n/((n*n+5)*lnn)
Исследовать на сходимость ряд
сумма от 2 до бесконечности n/((n*n+5)*lnn)
Обсуждение
Неизвестный
02.12.2007, 00:51
общий
это ответ
Здравствуйте, Shulga sergei!
Подобный вопрос обсуждался в 111470.
Ряд расходится. Я Вам приведу доказательство с использованием только элементарной математики путём повторного использования способа доказательства расходимости гармонического ряда ∑1/n.
Сначала напомню сам способ.
Веcь ряд мы разбиваем на частичные суммы от n=2<sup>k-1</sup>+1 до n=2<sup>k</sup>.
Каждый член суммы больше последнего 1/2<sup>k</sup>. Таких членов 2<sup>k-1</sup>. Частичная сумма больше (1/2<sup>k</sup>)2<sup>k-1</sup>=1/2.
Сумма k частичных сумм больше k/2, т.е. неограничена.
Теперь показываем как этот же принцип применим к нашему ряду.
Точно так же разбиваем на группы от n=2<sup>k-1</sup>+1 до n=2<sup>k</sup>.
Каждый член суммы больше последнего 2<sup>k</sup>/[(2<sup>2k</sup>+5)ln2<sup>k</sup>]>2<sup>k</sup>/[2<sup>2k+1</sup>kln(2)]=1/[2<sup>k+1</sup>kln2].
Таких членов 2<sup>k-1</sup>. Частичная сумма больше (1/[2<sup>k+1</sup>kln2])2<sup>k-1</sup>=1/(4kln2).
Таким образом, сумма k частичных сумм больше ∑1/(4nln2)=1/(4ln2)∑1/n, т.е. больше суммы гармонического ряда, который как мы уже доказали расходится.
Будут вопросы - пишите.
Подобный вопрос обсуждался в 111470.
Ряд расходится. Я Вам приведу доказательство с использованием только элементарной математики путём повторного использования способа доказательства расходимости гармонического ряда ∑1/n.
Сначала напомню сам способ.
Веcь ряд мы разбиваем на частичные суммы от n=2<sup>k-1</sup>+1 до n=2<sup>k</sup>.
Каждый член суммы больше последнего 1/2<sup>k</sup>. Таких членов 2<sup>k-1</sup>. Частичная сумма больше (1/2<sup>k</sup>)2<sup>k-1</sup>=1/2.
Сумма k частичных сумм больше k/2, т.е. неограничена.
Теперь показываем как этот же принцип применим к нашему ряду.
Точно так же разбиваем на группы от n=2<sup>k-1</sup>+1 до n=2<sup>k</sup>.
Каждый член суммы больше последнего 2<sup>k</sup>/[(2<sup>2k</sup>+5)ln2<sup>k</sup>]>2<sup>k</sup>/[2<sup>2k+1</sup>kln(2)]=1/[2<sup>k+1</sup>kln2].
Таких членов 2<sup>k-1</sup>. Частичная сумма больше (1/[2<sup>k+1</sup>kln2])2<sup>k-1</sup>=1/(4kln2).
Таким образом, сумма k частичных сумм больше ∑1/(4nln2)=1/(4ln2)∑1/n, т.е. больше суммы гармонического ряда, который как мы уже доказали расходится.
Будут вопросы - пишите.
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