How do I calculate the sum $$\sum_{n=0}^\infty\frac{(n+1)(n+2)}{2}x^n$$ I know this sum will be finite for $|x|<1$ and will diverge for other values of $x$. Since for other sums it was common to derivate in order to have a sum in the form $\sum_{n=0}^\infty x^n=\frac{1}{1-x}$ I thought it would be a good idea to integrate. However I sill can't solve it. Wolfram Alpha says the sum is $\frac{-1}{(x-1)^3}$. I would appreciate if someone could guide me to this result.
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https://math.stackexchange.com/questions/746388/calculating-1-frac13-frac1-cdot33-cdot6-frac1-cdot3-cdot53-cdot6-cdot – lab bhattacharjee Feb 07 '19 at 01:02
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1integrate term by term, twice. – Will Jagy Feb 07 '19 at 01:02
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https://math.stackexchange.com/questions/593996/how-to-prove-sum-n-0-infty-fracn22n-6/594019#594019 – lab bhattacharjee Feb 07 '19 at 01:02
2 Answers
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The hint:
Calculate $$\frac{1}{2}\left(\sum_{n=0}^{+\infty}x^{n+2}\right)''.$$
Michael Rozenberg
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Consider $f(x)=\sum_{n\geq 0}{x^{n+2}}$ and differentiate twice. Don’t forget to justify the differentiation under summation.
DINEDINE
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