Show that the series of functions $$\sum_{n=0}^\infty
\frac{n^2x}{1+n^4x^2}$$ converges pointwise, but not uniformly, on the interval $(0,1]$
This is the first time I have to show that a series of functions does not convere uniformly.
I managed to show that the series converges pointwise, using the comparison test.
However, I'm unsure how to prove it does not converge uniformly.
Since I don't know what the pointwise limit is, I suspect we have to use Cauchy's criterium:
$$\forall \epsilon > 0: \exists n_0: \forall p,q \geq n_0: \sup_{x \in (0,1]}\left|\sum_{k=0}^p f_k(x) – \sum_{k=0}^q f_k(x)\right| < \epsilon$$
But we should of course prove the negation of this criterium.
So, I started with seeing what I could do:
Let $p > q$, then I tried to estimate $$\sup_{x \in (0,1]}\left|\sum_{k=q+1}^p f_k(x) \right|$$
but couldn't find anything useful. Any clues?
Best Answer
The uniform convergence for a series of functions can also be stated as:
The logical negation of the above reads:
Then if we find some $x_m\in A$ such that $\left|\sum_{k=m}^\infty f_k(x_m)\right|\ge\epsilon$ for each $m\in\Bbb N$, for some fixed $\epsilon>0$, we are done.
A general procedure is try to find this $x_m$, to do this we can study what is $\sup_{x\in A}f_m(x)$. From a brief analysis you can find that $x_m:=1/m^2$ belongs to $A:=(0,1]$ and that $f_m(x_m)=1/2$.