Нужна помощь срочно!!!!!!!!. Created by kida445213. algebra-ru

Нужна помощь срочно!!!!!!!!...

kida445213 @kida445213

October 2021 1 11 Report

Нужна помощь срочно!!!!!!!!

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NNNLLL54

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$1)\; \; \sum \limits_{n=1}^{\infty }\dfrac{1}{n^2+5}\\\\\\\int\limits^{\infty }_1\, \dfrac{dx}{x^2+5}=\lim\limits _{A \to \infty}\int\limits^{A}_1\, \dfrac{dx}{x^2+5}=\lim\limits _{x \to \infty}\dfrac{1}{\sqrt5}\, arctg\dfrac{x}{\sqrt5}\Big|_1^{A}=\\\\\\=\lim\limits _{x \to \infty}\dfrac{1}{\sqrt5}\Big(arctg\dfrac{A}{\sqrt5}-arctg\dfrac{1}{\sqrt5}\Big)=\dfrac{1}{\sqrt5}\Big (\dfrac{\pi}{2}-arctg\dfrac{1}{\sqrt5} \Big)=const$

Несобственный интеграл сходится, значит и ряд сходится .

$2)\; \; \sum \limits_{n=1}^{\infty }\dfrac{n^2}{n^2+5}\\\\Neobxodimuj\; priznak:\; \; \lim\limits _{n \to \infty} a_n=\lim\limits _{n \to \infty}\dfrac{n^2}{n^2+5} =1\ne 0\; \to \; rasxoditsya\\\\\\3)\; \; \sum \limits_{n=1}^{\infty }\dfrac{n+1}{2n+5}\\\\Neobxodimuj\; priznak:\; \; lim\limits _{n \to \infty}\, a_{n}=\lim\limits _{n \to \infty}\dfrac{n+1}{2n+5} =\frac{1}{2}\ne 1\; \; \to \; \; \; rasxoditsya$

2 votes Thanks 1

Winner1919 Доброго времени суток! Вы умеете такое решать? https://znanija.com/task/37041536

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