Normal convergence?

Question

I am asked to show that the following series of functions does not converge normally.

$\sum_{i=1}^\infty \frac{1}{n} - \frac{1}{x+n} $ I showed that it converges simply (over $\mathbb{R}^+\cup 0$) and tried to show its uniform convergence through that of the series’ remainder, but I fell short.

Any help is greatly appreciated!

Answer 1

I think you want to evaluate the maximum value of $[\frac{1}{n} -\frac{1}{(n+x)}]$ over $\mathbb R^{+}$.

It is $\frac{1}{n}$.

Thus the series is not normally convergent.