TITLE: Calculate limit
QUESTION [5 upvotes]: I need to calculate limit 
$$\lim\limits_{n \to \infty} n^2 q^n,$$
where $|q|<1$. Any hints how to do that would be appreciated.

REPLY [2 votes]: Another fancy way: take the series
$$\sum_{n=1}^\infty n^2q^n\;,\;\;|q|<1$$
Applying the $\;n$-th root test we get:
$$\sqrt[n]{n^2|q|^n}=\sqrt[n]{n^2}\;|q|\xrightarrow[n\to\infty]{}|q|<1$$
and thus the series converges (even absolutely), and from here we get at once that
$$n^2q^n\xrightarrow[n\to\infty]{}0$$