Answer to Question #345754 in Real Analysis for Roy

Question

Check whether the series {Summation} n²x⁵/(n⁴+x³) , x belongs to [0, a] is uniformly convergent or not ,where a belongs to R

Answer 1

ANSWER . The series $\sum_{n=1}^{\infty}\frac{n^{2}x^{5}}{n^{4}+x^{3}}$ converges uniformly on $[0,a]$ .

EXPLANATION.

Since $a>0$ , then for all $x\in [0,a] , n\geq 1$ the inequality

$0\leq \frac{n^{2}x^{5}}{n^{4}+x^{3}}\leq \frac{n^{2}a^{5}}{n^{4} }=\frac {a^{5} }{n^{2}}$

is true. The series $\sum_{n=1}^{\infty}\frac{ a^{5}}{n^{2} }=a^{5}\cdot \sum_{n=1}^{\infty}\frac{ 1}{n^{2} }$ converges, because the series $\sum_{n=1}^{\infty}\frac{ 1}{n^{2} }$ converges ( p-series with p=2). So, by the Weierstrass M test the series $\sum_{n=1}^{\infty}\frac{n^{2}x^{5}}{n^{4}+x^{3}}$ converges uniformly on $[0,a]$ .