Question

Show that the series , ∞Σ n=1 (x/1+n^2x^2) is uniformly convergent in [ k,1] for any k>0.

Answer 1

If $x\in[k,1]$ , $0<k<1$ , then $\frac{x}{1+n^2x^2}\leq \frac{x}{n^2x^2}=\frac{1}{n^2x}\leq \frac{1}{n^2 k}$.

The expression on the right hand does not depend on x and the series $\sum\limits_{n=1}^{\infty}\frac{1}{n^2 k}$ is convergent, therefore the initial series is uniformly convergent on the segment [k,1].

In order to prove that the series is uniformly convergent for any $k>0$ , it is sufficient to show that for any $N\in\mathbb{N}$ there exists $x\in[0,1]$ , $q> p> N$ such that $\sum\limits_{n=p}^{q}\frac{x}{1+n^2x^2}\geq 1/5$ .

Put x=1/N, p=N+1, q=2N. Then $nx\leq 2$ for all $n=p, p+1, \dots, q$ and

$\sum\limits_{n=p}^{q}\frac{x}{1+n^2x^2}\geq \sum\limits_{n=p}^{q}\frac{x}{1+2^2}=\sum\limits_{n=N+1}^{2N}\frac{1/N}{5}=1/5$

Therefore, the series $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is uniformly convergent for any $k>0$ .