How can you check if a series$ ∑_1^{\infty} x^{2}e^{−nx}$ is convergent when $ x \in [0,+∞)$?

Question

How can you check if a series $$∑_1^{\infty} x^2e^{−nx}$$ is convergent when $x$ belongs to $[0,∞)$ step by step? Can D'Alembert's theorem be used? Could someone write it step by step?

Answer 1

For $x=0$ we have $\sum _{k=1}^N x^2 e^{-kx} = 0$ for every $N$, so the series converges (to $0$). Otherwise, as pointed out in the comments, we have $$ \sum x^2 e^{-nx} = x^2 \sum e^{-nx} = x^2 \sum \left ( e^{-x} \right )^n. $$ This series converges if and only if the geometric series $\sum (e^{-x})^n$ converges. This occurs if and only if $e^{-x}< 1$ (i.e $x>0$).