Pointwise convergence of series of functions implies uniform convergence? In my Real Analysis class, a question came up, but I can't come up with a proof or disproof. Here's the conjecture:
If $\sum f_n \rightarrow f$ pointwise on $(a, b) \subseteq \mathbb{R}$ then $\sum f_n \rightarrow f$ converges uniformly on $[c, d] \subseteq (a, b)$.
I'm convinced that divergence may occur at the endpoints and having $[c, d]$ gives us a better interval to work with by getting rid of the bad stuff. To be more specific, since $\sum f_n \rightarrow f$ pointwise, the number series $\sum f_n(x)$ converges $\forall x \in (a, b) \implies f_n(x) \rightarrow 0$. But, we can't come up with a number sequence $(x_n)$ where each $x_n \in [c, d]$ such that $f_n(x_n) \nrightarrow 0$ since we got rid of our endpoints. 
Is my conjecture valid? If so, how would I prove or disprove my conjecture?
 A: Let $a=-1, b=1, c=-\frac 1  2, d=\frac 1  2$. Let $g_n(x)=n|x|$ for $|x| \leq \frac 1  n$  and $1$ for $|x| > \frac 1  n$. Let $g(x)=1$ for $x \neq 0$ and $g(0)=0$.  Then $g_n \to g$ pointwise in $(a,b)$ but not uniformly in $[c,d]$. Now take $f_1=g_1, f_2=g_2-g_1,f_3=g_3-g_2,...$. 
A: A counterexample is the series with partial sums
$$S_n(x) = xe^{-x^2}+\sum_{k=1}^{n-1} \left[(k+1)x e^{-(k+1)^2 x^2} - kxe^{-k^2x^2}\right] = nxe^{-n^2x^2},$$
where $S_n(x) \to S(x) =0$ as $n \to \infty$ pointwise for all $x \in \mathbb{R}$.  However, convergence is not uniform on $[0,1]$ since $|S_n(1/n) - S(1/n)| = e^{-1} \not \to 0$.
A: Since the counterexample is already given (by Kavi), I'd like to comment about pointwise
and uniform convergence.
First, remember that a series is just a sequence (of partial sums) and a sequence can be 
written as a series as Kavi's answer shows. So, we'll talk only about sequences.
Pointwise convergence means: for each individual $x$, the sequence $f_n(x)$ converges.
This means that once you've picked $x$ you can tell if the sequence $f_n(x)$ is close
to its limit. Different $x$'s may converge at different rates (I mean their respective 
sequences $(f_n(x))_n$).
You cannot know if the function sequence $(f_n)_n$ itself is close to the
(pointwise) limiting function. That's what the notion of uniform limit is for...
Uniform convergence means: Now you can tell if the whole function sequence $(f_n)_n$ is
close to the (pointwise) limiting function. By this I mean comparing their graphs. From
some $N\in\mathbb{N}$ on, the graph of $f_n$ is really close to the graph of its limit for
$n\geq N$.
