Определить сходимость ряда ((x+4)/(x+3))^((2x-1)/x) ОЧЕНЬ ВАЖНО, ПОМОГИТЕ!!!. Created by Tekari. matematika-ru

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Tekari @Tekari

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Определить сходимость ряда ((x+4)/(x+3))^((2x-1)/x)

ОЧЕНЬ ВАЖНО, ПОМОГИТЕ!!!

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NNNLLL54

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$\sum\limits _1^{\infty }\Big ( \frac{n+4}{n+3} \Big )^{ 2n-1}=\sum \limits _1^{\infty }\Big (1+ \frac{1}{n+3}\Big )^{2n-1}$

Необходимый признак сходимости:

$\lim\limits _{n\to \infty }a_{n}= \lim\limits _{n \to \infty} \Big (1+ \frac{1}{n+3}\Big )^{2n-1} = \lim\limits _{n \to \infty} \Big (1+ \frac{1}{n+3}\Big )^{(n+3)\cdot \frac{2n-1}{n+3}} =\\\\= \lim\limits _{n \to \infty} \Big (\Big (\underbrace {1+ \frac{1}{n+3}\Big )^{n+3}}_{\to e}\Big )^{\frac{2n-1}{n+3}}=e^{ \lim\limits _{n \to \infty} \frac{2n-1}{n+3}} =e^2\ne 0\; \; \to \; \; rasxoditsya$

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