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].