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Need a hint to compute $\displaystyle \sum_{x=0}^\infty xa^x$ and $\displaystyle \sum_{x=0}^\infty x^2a^x$, where $a \in (0,1)$.

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  • $\begingroup$ This will help: math.stackexchange.com/questions/647587/… $\endgroup$
    – Brenton
    May 31, 2015 at 18:39
  • $\begingroup$ Why that strange way to write the limits in the sum? $\endgroup$
    – AdLibitum
    May 31, 2015 at 18:40
  • $\begingroup$ I've posted answers to this question several times. I wonder if it's our most frequently re-posted question. $\endgroup$ May 31, 2015 at 18:56
  • $\begingroup$ @MichaelHardy: He seems to be the first to also ask about $\sum \limits _{n=0} ^\infty n^2 a^n$. $\endgroup$
    – Alex M.
    May 31, 2015 at 19:00
  • $\begingroup$ @Brenton: Apparently, the original is math.stackexchange.com/questions/67364/sequence-sum-question $\endgroup$
    – Alex M.
    May 31, 2015 at 19:01

2 Answers 2

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Hint: Differentiate $\sum_{x=0}^{\infty}a^x$ with respect to $a$.

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Consider the genral term and rewrite as $$ia^i=a(ia^{i-1})=a\frac{d}{da}(a^i)$$ So we can write the series as $$a\frac{d}{da}\sum_{i=0}^{\infty}a^i=a\frac{d}{da}\frac{1}{1-a}=\frac{a^2}{(1-a)^2}$$ I leave the second one to you. Hope this helps.

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