Show $\displaystyle \sum_{n=1}^{\infty} e^{-nx}x^n$ converges uniformly for $0 \leq x \leq 10$.
Hint: Find the maximum of $xe^{-x}$ on the interval.
I was able to find the maximum at $x=1$ which is $\displaystyle \frac{1}{e}$. I think I can use the Weierstrass M test for this.
That is, I need to construct a sequence of positive numbers $M_1, M_2, ...$ s.t. $|e^{-kx}x^k| \leq M_k, \forall x \in [0,10]$
and
$\displaystyle \sum_{n=1}^{\infty}M_n \lt \infty$.
Now the bit I am struggling with is coming up with a sequence of $M$'s s.t. their sum is convergent.
I also see that $\displaystyle \frac{1}{e} \lt 1$ and $\displaystyle \frac{1}{e^n}$ converges to $0$. But I am struggling with using this fact to constructing the $M$'s.
I would love to have some hints please.
Thank you.