Monday 25 August 2014

sequences and series - How to compute $sum_{x=0}^{infty}xa^x$?





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


Answer



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


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