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Can I evaluate: $\sum_{n=0}^{5}{nx^n}$ to get something nicer?

I know I can evaluate:

$\sum_{n=0}^{N-1}{x^n} = (1-x^N)(1-x)^{-1}$ for $x \neq 1$

can I do something similar with the first sum?

Nillo
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  • These posts might be useful: http://math.stackexchange.com/questions/87030/proof-by-induction-sum-limits-i-0n-i-2i-1-n1-2n-1 http://math.stackexchange.com/questions/11464/how-to-compute-the-formula-sum-r-1d-r-cdot-2r (Notice that some answers answre a more general question then asked in the title of that question.) http://math.stackexchange.com/questions/90637/what-is-the-limit-of-sum-limits-n-1-inftyn2-3n http://math.stackexchange.com/questions/30732/how-can-i-evaluate-sum-n-0-infty-n1xn – Martin Sleziak Oct 23 '15 at 13:50

1 Answers1

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$\textbf{hint}$

$$ x\sum \dfrac{d}{dx}x^{n} = x\dfrac{d}{dx}\sum x^n = \sum nx^n $$

Chinny84
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  • So I get: $\frac{Nx^2}{1-x} + \frac{x^N-1}{(1-x)^2}$ ? – Nillo Oct 23 '15 at 13:30
  • It does not look right to me (unless there are some typos). Try putting in some numbers for $x$ and $N$ and compute the series by hand and then by formula. – Chinny84 Oct 23 '15 at 13:38
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    You were right lots of typos actually.. I recalculated and got $\frac{Nx^N}{x-1} + \frac{x^{N+1}-x}{(x-1)^2}$ It agrees with the sum :D – Nillo Oct 23 '15 at 13:46
  • np. Glad it helped you. Remember these neat little tricks in the hint above, as they are always helpful at some point. – Chinny84 Oct 23 '15 at 13:48