# How do I evaluate the power series \sum_{n=1}^\infty\frac{n}{9^n} without using the formula

How do I evaluate the power series
$\sum _{n=1}^{\mathrm{\infty }}\frac{n}{{9}^{n}}$
without using the formula for infinite geometric series?
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poleglit3
Check with induction that the partial sum is
$\sum _{k=1}^{N}\frac{k}{{9}^{k}}=\frac{9-{9}^{-N}\left(8N+9\right)}{64}$
Evaluate the limit when N goes to infinity and you get the result.
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Beverly Smith
Notice that $x\frac{d}{dx}\frac{1}{1-x}=\sum _{i=0}^{\mathrm{\infty }}i{x}^{i}$. So differenciating you get
$x\frac{d}{dx}\frac{1}{1-x}=\frac{x}{{\left(1-x\right)}^{2}}$
Substituting $x=\frac{1}{9}$ you get required answer.
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karton

Denote . You have the closed system

which gives the value of A. Of course, as a byproduct of the computation one also gets the value of the geometric series C.