I have to show that the series $\sum^\infty_{k=1}kx^k$ converges uniformly on the interval $\left[-a,a\right]$ when $0<a<1$ but not on the interval $(-1,1)$.
What I have gotten so far:
The series looks like a geometric series and I think I can use that information and potentially use Weierstrass M-test but I don't know how.
Best Answer
Hint
If $a\in (0,1)$ and $x\in (-a,a)$, then $$|kx^k|\leq k|a|^k,$$ and thus the series converges normally.
Now, $$\sup_{x\in(0,1)}\left|\sum_{k=n+1}^\infty kx^k\right|\geq\left|\sum_{k=n+1}^\infty k\left(1-\frac{1}{m}\right)^k\right|\geq(n+1)\left(1-\frac{1}{m}\right)^{n+1}\sum_{k=0}^\infty \left(1-\frac{1}{m}\right)^k$$ $$=(n+1)\left(1-\frac{1}{m}\right)^{n+1}m\underset{m\to \infty }{\longrightarrow }\infty .$$