I am trying to understand Tauber's Theorem:
Part of the theorem involves this assumption:
But since k|ak| → 0, it follows that 1/n (summation from k=1 to n) k|ak| -> 0.
*Here, ak is a sequence in R.
What I want to know is how we can make the assumption:
|ak| → 0, then 1/n (summation from k=1 to n) k|ak| -> 0. Does anyone know a proof?
Recall that If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $. This explains the statement you quoted:
since $k|a_k| \to 0$, it follows that $\frac{1}{n} \sum_{k=1}^n k|a_k| \to 0$
Having only $|a_k|\to 0$ would not be enough to conclude $\frac{1}{n} \sum_{k=1}^n k|a_k| \to 0$. Indeed, take $a_k = 1/k$: then $$\frac{1}{n} \sum_{k=1}^n k|a_k| = 1 \not\to 0$$ Or take $a_k=1/\sqrt{k}$ to make it even worse.
