Answer to Question #304925 in Real Analysis for Sarita bartwal

Question

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Answer 1

We test the uniform convergence of series only for $x>0$. For $x<0$ equations $n^5+x^3=0$ have solutions for $n\in\mathbb{N}$ and therefore some terms under the sum will be equal to infinity. The terms $\frac{1}{n^5+x^3}$ can be bounded in the following way: $|\frac{1}{n^5+x^3}|<\frac{1}{n^5}$. Series $\sum_{1}^{+\infty}\frac{1}{n^5}$ converge due to the integral criterion. Namely, $\int_{1}^{+\infty}\frac{dx}{x^5}=-\frac{1}{4}x^{-4}|_{1}^{+\infty}=\frac{1}{4}$ The original series $\sum_{1}^{+\infty}\frac{1}{n^5+x^3}$ converge uniformly due to the Weierstrass criterion.