If |x|<1 show that:
$\sum_{k=1}^∞ kx^k = \frac{x}{(1-x)^2}$
I know that I should use partial fraction expansion. But don't really understand how to do it.
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Factor one $x$ out of both sides. Integrate what remains. If you really want partial fraction expansion, you know you can write $$\frac{x}{(1-x)^2}=\frac{a}{1-x}+\frac{b}{(1-x)^2}$$ Then identify $a$ and $b$. – Jean-Claude Arbaut Nov 30 '22 at 21:33
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Other hint: differentiate the power series for $\frac{1}{1-x}.$ – Sean Roberson Nov 30 '22 at 21:38