A question on the uniform convergence of a series

Question

Consider the series $$\sum_{k=0}^{\infty}e^{-kx}kx$$ for $x\geq 0$.

I have managed to show that this series converges pointwise for all $x\geq0$, and I am currently trying to prove that this series does not converge uniformly for $x\geq0$. However I am kind of stuck. Any hint will be appreciated. Thanks.

Answer 1

If $\sum_k f_k$ converges uniformly then $\Vert f_k\Vert_\infty:=\sup_x|f_k(x)|$ converges to $0$ for $k\to\infty$. Now just notice that $\sup_{x\geq0}e^{-kx}kx=\sup_{x\geq0}e^{-x}x$ is $>0$ and independent of $k$, so it doesn't converge to $0$.

Answer 2

hint: we can prove that cauchy criterion does not hold, we need to show that there exist $\epsilon > 0$ such that for every $N>0$ there exists $m\ge n\ge N$ and $x \ge 0$ such that $\sum_{k=n}^{m}e^{-kx}kx \ge \epsilon$. and indeed let $\epsilon = e^{-1}$ and let $N>0$ try looking at $x = 1/N$