I'm currently working on proving some theorems and there is one recurring problem that I somehow can't solve. $a_n$ is a real sequence in either $[0,1]$ or $\mathbb{R}$ that approaches $0$.
$$\lim\limits_{n\to\infty}\sum\limits_{k=1}^\infty (a_{n})^kn = \sum\limits_{k=1}^\infty \lim\limits_{n\to\infty} (a_{n})^kn$$ Where $$na_n\underset{n\to\infty}{\longrightarrow}C \in [0,\infty)$$ therefore $$na_n^k\underset{n\to\infty}{\longrightarrow} 0 \text{, for k>1}$$ I wanted to switch the limits and summation using the dominated convergence theorem, but failed at finding a limiting sequence because I don't know the specific form of $a_n$. Am I missing something obvious?
I think this trick will work:
Choose $N$ such that for $n>N$ $n \cdot a_n<(C+1)$, and $|a_n|<1/2$ (since $a_n$ goes to $0$ we can find such a $N$).
Then, $|na_n^k| \leq (C+1)(1/2)^{k-1}$- this shall be your dominating sequence.