Find the value that $\sum_{j\geqslant 1} j^k*2^{-j}$ converges to for $k \in {N}$.

Question

Context: I am trying to compute the expected value of a random variable $X$ whose PMF is given by $P_X(x)=(\tfrac12)^x$. This results in verifying if $\sum_{x\in{N}}x2^{-x}$ converges, and if so, to what value. I used the absolute convergence test and verified that this series converges. I also know from checking online calculators that this series converges to $2$.

Further, I need to find the variance of $X$ as well. I'm going about that by computing $E[X^2]$ which means I need to evaluate $\sum_{x\in{N}}x^2*2^{-x}$. I have no idea how to calculate that either, although I have found from the ratio test that this series also converges.

So, the question is, how to find the value for $\sum_{x\in{N}}x^k*2^{-x}$ for $k = 1,2$ and is there a formula for general $k\in{N}$.

@J.W.Tanner thank you, any ideas on the similar looking series at the end? Updated the question — Alborz, Aug 18 '23 at 16:29
Do you know that you can compute $\mathbb EX$ as $G_X'(1)$ where $G_X(x) = \sum_{k\geqslant 0} \mathbb P(X=k)z^k$? In this case, you can compute that $G_X(x) = 2/(2-x)$ and continue with the computation of the expected value. — Pedro, Aug 18 '23 at 16:30
@Alborz For the variance, you can check it is $G_X'(1) +G_X''(1) - G_X'(1)^2$. — Pedro, Aug 18 '23 at 16:32
The standard approach is to find the value for a sum of a finite number of terms as a function of the number of terms, and then take the limit. — RicardoMM, Aug 18 '23 at 16:36
@AnneBauval This does not answer the follow up question regarding series $x^2*2^{-x}$ — Alborz, Aug 18 '23 at 17:05
@Pedro MGF is definitely the way to go here. Thanks so much. If you'd like to post as an answer I will accept. — Alborz, Aug 18 '23 at 17:07
@Alborz Please focus on one question per post. Your main question was a duplicate and your "Follow up question" should be another post (certainly as well a mega-duplicate). — Anne Bauval, Aug 18 '23 at 17:08
For this now more general question, see General formula for $\sum_{n=1}^{\infty} n^k r^n$ for arbitrary $k$ and https://en.wikipedia.org/wiki/Polylogarithm — Anne Bauval, Aug 18 '23 at 17:25
@MarianoSuárez-Álvarez before editing he had a geometric series, which works with that, and I meant for any geometric series, not any series — RicardoMM, Aug 18 '23 at 17:39

Find the value that $\sum_{j\geqslant 1} j^k*2^{-j}$ converges to for $k \in {N}$.

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