Uniform (but not normal) convergence of a series of function

Question

consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$. It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal convergence. I want to know whether there is uniform convergence towards $f$ or not.

I tried to disprove uniform convergence by looking at $|\sum_{k=0}^n f_k(n)-f(n)|$ : the sum is unbounded and equivalent to $\ln n$. The problem is that I don't know how to estimate $f(n)$.

I'm sure I'm missing something not too difficult here, so help would be appreciated.

Answer 1

Let us consider only positive $x$, then all $f_k(x)$ are positive, and we have

$$\left\lvert\sum_{k=1}^n f_k(x) - f(x) \right\rvert = f(x) - \sum_{k=1}^n f_k(x) = \sum_{k=n+1}^\infty f_k(x).$$

Now let's choose a suitable $x$, say $x = 2n$. Then

$$f(2n) - \sum_{k=1}^n f_k(2n) > \sum_{k=n+1}^{2n} f_k(2n) > n\cdot f_{2n}(2n) = \frac14,$$

so the convergence is not uniform.

Answer 2

Hint find where $f_n'(x)=0$, look at $x=n$, $f_n(x)$ has maximum, and $M_n={1\over 2n}$ as $\sum {1\over 2n}$ is divergent so your series is not uniformly convergent.