For $x\in\mathbb{R}$ with $|x|<1$. Find the value of
$$\sum_{n=0}^{\infty} (1+n)x^n$$
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2Use integration (if you're allowed). – Zardo Jun 12 '15 at 15:47
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it should be solved without calculus use as the problem is from non calculus based curriculum. – gaufler Jun 12 '15 at 15:50
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$$\sum_{n=0}^{\infty} (1+n)x^n=\sum_{n=0}^{\infty}x^n+\sum_{n=0}^{\infty}nx^n$$ $$\frac{1}{1-x}+x\frac{d}{dx}(\frac{1}{1-x})=\frac{1}{1-x}+\frac{x}{(1-x)^2}=\frac{1}{(1-x)^2}$$
E.H.E
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Write $$\sum_{n \geq 0 }(n+1)x^n =\sum_{n \geq 0 }nx^n + \sum_{n \geq 0 }x^n $$
And observe that$$ \sum_{n \geq 0 }nx^n =\sum_{n \geq 0 }x\frac{d}{dx}x^n= x\frac{d}{dx}\sum_{n \geq 0 }x^n$$
Can you continue from here?
Joaquin Liniado
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1I find it interesting that OP said above to not use calculus, and accepted your answer using a derivative. – A. Thomas Yerger Jun 12 '15 at 16:11