Necessary Condition for uniform convergence of series of functions

Question

I would like to make sure that the follwing is a necessary condition for uniform convergence of series of functions: :Let $$ \sum _{n=1}^{\infty }f_{n}(x) $$ be a series of functions, than a necessary condition for its uniform convergence is that $$ \left \{ f_{n}(x) \right \}_{n=1}^{_{n=\infty }} $$ uniformly converges to zero.

Is that true?

Answer 1

Note that

$$|f_m(x)| =\left|\sum_{k=n+1}^m f_k(x) - \sum_{k=n+1}^{m-1} f_k(x)\right|\leq \left|\sum_{k=n+1}^m f_k(x)\right|+\left|\sum_{k=n+1}^{m-1} f_k(x)\right|.$$

Suppose the series converges uniformly on $D$.

For every $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that if $m-1 \geq n \geq N$ then for every $x \in D$

$$\left|\sum_{k=n+1}^m f_k(x)\right| < \frac{\epsilon}{2}$$

and

$$\left|\sum_{k=n+1}^{m-1} f_k(x)\right| < \frac{\epsilon}{2}.$$

Hence, if $m > N+1$ then for every $x \in D$

$$|f_m(x)| < \epsilon,$$

and $(f_m)$ converges uniformly to $0$.

Answer 2

The uniform convegence of $$ \sum f_n(x) $$ is in fact the convergence of the series $$ \sum f_n $$ in a the functional space with the norm $\|.\|_\infty$ (called: "of uniform convergence").

So as in every normed space, convergence of the series implies $$ \|f_n\|_\infty \to 0 $$