If $\sum a_n$ is a convergent series of real numbers prove that the series:
1) $\sum a_n e^{-nx}$ is uniformly convergent on $[0,\infty)$;
2) $\sum \frac{a_n}{n^x}$ is uniformly convergent on $[0,1]$.
My Try: 1) Let $v_n = e^{-nx}$. For each $x \in [0,\infty)$, the sequence $v_n$ is monotonic and for all $x \in [0,\infty)$, $|v_n(x)| = \frac{1}{e^{nx}} \leq 1$ for all $n \in \mathbb N$.
And $\sum a_n$ is a convergent series of real numbers. Thus by Abel's Test we have $\sum a_n e^{-nx}$ is uniformly convergent on $[0,\infty)$.
Similar conditions holds for 2. Is the proof correct??