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Консультация № 115738
24.12.2007, 14:56
50.00 руб.
0
1
1
исследовать на сходимость ряд:
сумма от n=1 до беск. 2/([5^(n-1)]+n-1)
сумма от n=1 до беск. 2/([5^(n-1)]+n-1)
Обсуждение
Неизвестный
24.12.2007, 20:53
общий
это ответ
Здравствуйте, Татьянка!
Для проверки на сходимость используем признак Даламбера.
Найдем границу lim<sub>n→∞</sub>U<sub>n+1</sub>/U<sub>n</sub>
lim<sub>n→∞</sub>U<sub>n+1</sub>/U<sub>n</sub> = lim<sub>n→∞</sub>2/(5<sup>n</sup>+n)/2/(5<sup>n-1</sup>+n-1) =
= lim<sub>n→∞</sub>(5<sup>n-1</sup>+n-1)/(5<sup>n</sup>+n) = lim<sub>n→∞</sub>5<sup>n-1</sup>·(1+(n-1)/5<sup>n-1</sup>)/(5<sup>n-1</sup>·(5+n/5<sup>n-1</sup>)) = {сокращаем на 5<sup>n-1</sup>} = lim<sub>n→∞</sub> (1+(n-1)/5<sup>n-1</sup>)/(5+n/5<sup>n-1</sup>) = [обращаю Ваше внимание, что lim<sub>n→∞</sub>n/a<sup>n</sup> = 0] = (1+0)/(5+0) = 1/5.
Так как <font color=blue>1/5 < 1</font> следовательно <b>ряд сходится</b>.
Good Luck!!!
Для проверки на сходимость используем признак Даламбера.
Найдем границу lim<sub>n→∞</sub>U<sub>n+1</sub>/U<sub>n</sub>
lim<sub>n→∞</sub>U<sub>n+1</sub>/U<sub>n</sub> = lim<sub>n→∞</sub>2/(5<sup>n</sup>+n)/2/(5<sup>n-1</sup>+n-1) =
= lim<sub>n→∞</sub>(5<sup>n-1</sup>+n-1)/(5<sup>n</sup>+n) = lim<sub>n→∞</sub>5<sup>n-1</sup>·(1+(n-1)/5<sup>n-1</sup>)/(5<sup>n-1</sup>·(5+n/5<sup>n-1</sup>)) = {сокращаем на 5<sup>n-1</sup>} = lim<sub>n→∞</sub> (1+(n-1)/5<sup>n-1</sup>)/(5+n/5<sup>n-1</sup>) = [обращаю Ваше внимание, что lim<sub>n→∞</sub>n/a<sup>n</sup> = 0] = (1+0)/(5+0) = 1/5.
Так как <font color=blue>1/5 < 1</font> следовательно <b>ряд сходится</b>.
Good Luck!!!
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