uniform convergence series

**problem** · January 7th 2010, 07:10 PM

Suppose $\sum f_n(x)$ converges uniformly on $A$ and $g :A \rightarrow B$ is a continuous function.
Do we have any conclusion about the series $\sum g(f_n(x))$ ?
( $A$ and $B$ are subsets of real number)

**putnam120** · January 7th 2010, 08:49 PM

Originally Posted by problem

Suppose $\sum f_n(x)$ converges uniformly on $A$ and $g :A \rightarrow B$ is a continuous function.
Do we have any conclusion about the series $\sum g(f_n(x))$ ?
( $A$ and $B$ are subsets of real number)

No.

$\sum g(f_n(x))$ might diverge or converge. Just take $f_n(x)=2^{-n}$ and $g(x)=1$ , in this case we have divergence. However, if $g(x)=x$ we have convergence. So we need to know more about $g$

**problem** · January 8th 2010, 08:19 AM

putnam120,if I change the g to be a uniformly continuous function,can I claim that \sum g(f_n) converges uniformly?

Or it will be the case that convergence/uniform convergence of a series is not invariant under any continuous/uniform continuous function?

**Shanks** · January 8th 2010, 08:49 AM

Notice that g(x)=1 and g(x)=x are both uniformly continious function, the claim still can't be true.

**problem** · January 8th 2010, 08:55 AM

So is there any properties that a function should have so that the uniform convergence of a series is invariant under the particular function?

**Shanks** · January 8th 2010, 08:59 AM

I think, the answer is negative.

**Jose27** · January 8th 2010, 03:42 PM

Originally Posted by problem

putnam120,if I change the g to be a uniformly continuous function,can I claim that \sum g(f_n) converges uniformly?

Or it will be the case that convergence/uniform convergence of a series is not invariant under any continuous/uniform continuous function?

Going by what I think you mean you're misunderstanding your question:

If $f_n : A\rightarrow B$ and $g: B \rightarrow \mathbb{R}$ are such that $f_n \rightarrow f$ unif. on $A$ and $g$ is unif. cont. on $B$ then $g(f_n)\rightarrow g(f)$ unif. on $A$ . Applying this result to the partial sums of a series, say $\sum_{k=1}^{\infty } h_k = h$ (which converges unif.) we get that $g(\sum_{k=1}^{n} h_k) \rightarrow g(h)$ uniformly. Thus unif. continuity does preserve unif. convergence. What you're asking however is convergence of the series given by $\sum_{k=1}^{\infty } g(h_k)$ . Do you see the difference?

PS. I think this works for your original question:

Let $h_k,h : A \rightarrow B$ such that $\sum_{k=1}^{\infty } h_k(x) = h(x)$ be a unif. conv. series of functions such that it converges absolutely for all $x \in A$ , and $g: B\rightarrow \mathbb{R}$ be unif. continous and $\vert g(x) \vert \leq M \vert x \vert$ for all $x\in B$ then $\sum_{k=1}^{\infty } g(h_k)$ converges unif. on $A$ and absolutely for all $x\in A$

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