Answer to Question #319201 in Calculus for Mani

Question

Check whether the series sum_(n=1)^(oo)(nx)/(n^(4)+x^(3)) x in 0 alpha is uniformly convergent or not

Answer 1

ANSWER The series $\sum_{n=1}^{\infty}\frac{nx}{n^{4}+x^{3}}$ is uniformly convergent on $\left [ 0, \alpha \right ]$ .

EXPLANATION.

Since a) $0\leq nx\leq n \alpha$

b) $n^{4}+x^{3}\geq n^{4}$

for $x\in[0,\alpha]$ and $n\geq1$ , then

$0\leq\frac{nx}{n^{4}+^x{3}}\leq\frac{n\alpha}{n^{4}}=\frac{\alpha}{n^{3}}$ .

The series $\sum_{n=1}^{\infty}\frac{\alpha}{n^{3}}=\alpha\sum_{n=1}^{\infty }\frac{1}{n^{3}}$ is convergent , because the series $\sum_{n=1}^{\infty }\frac{1}{n^{3}}$ is a $p$-series for $p=3$ .

Therefore, by the Weierstrass M-Test the series $\sum_{n=1}^{\infty}\frac{nx}{n^{4}+x^{3}}$ is uniformly convergent on $\left [ 0, \alpha \right ]$ .