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I need to write the series

$$\sum_{n=0}^N nx^n$$

in a form that does not involve the summation notation, for example $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$. Does anyone have any idea how to do this? I've attempted multiple ways including using generating functions however no luck

Jez
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2 Answers2

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Hint:

First of all, we have that $$\sum_{n=0}^{N}x^n=\frac{x^{N+1}-1}{x-1}$$ for $x\neq1$. So $$\begin{align} \sum_{n=0}^{N}nx^n&=x\sum_{n=0}^{N}nx^{n-1}\\ &=x\sum_{n=0}^{N}\frac{d}{dx}x^{n}\\ &=x\frac{d}{dx}\sum_{n=0}^{N}x^{n}\\ &=x\frac{d}{dx}\frac{x^{N+1}-1}{x-1}\\ \end{align}$$ Can you take it from here?

clathratus
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You can use the fact that\begin{align}\sum_{n=0}^Nnx^n&=x\sum_{n=0}^Nnx^{n-1}\\&=x\left(\sum_{n=0}^Nx^n\right)'\\&=x\left(\frac{1-x^{N+1}}{1-x}\right)'.\end{align}

José Carlos Santos
  • 427,504