I'm a bit rusty on series summation. What are some basic techniques to evaluate a summation thats not given in a formula?

For example, how do I evaluate $\sum\limits_{k=1}^\infty k(\frac{10}{11})^{k-1}$

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    $\begingroup$ Consider $\sum_{k=1}^\infty kx^{k-1}$. $\endgroup$ – Lord Shark the Unknown Jul 19 '17 at 17:36
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    $\begingroup$ ... That is $\frac{d}{dx}\sum_{k\geq 1}x^k = \frac{d}{dx}\frac{x}{1-x}=\frac{1}{(1-x)^2}$ for any $x\in(-1,1)$. $\endgroup$ – Jack D'Aurizio Jul 19 '17 at 17:39

Using calculus $$ \sum_{k=1}^{\infty} k q^{k-1} = \frac{\partial}{\partial q}\sum_{k=0}^{\infty} q^k = \frac{\partial}{\partial q}\frac{1}{1-q} = \frac{1}{(1-q)^2}. $$ Using results from probability: recall that if $X\sim Geo(p)$, then $P(X=k)=(1-p)^{k-1}p$, and $EX = 1/p$, namely, $$ EX = \sum_{k=1}^{\infty}k(1-p)^{k-1}p = \sum_{k=1}^{\infty}kq^{k-1}p = p\sum_{k=1}^{\infty}kq^{k-1}=\frac{1}{p}, $$ i.e., $$ \sum_{k=1}^{\infty}kq^{k-1} = \frac{1}{p^2} = \frac{1}{(1-q)^2}. $$


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