<s>[INST] A simple series $\sum_{i=1}^\infty \frac{i}{2^i} = 2$ I don't do math a long time, so I completely don't remember how to prove that:
$$
\sum_{i=1}^\infty \frac{i}{2^i} = 2
$$
Can anybody help me?
 [/INST] $S = x + 2x^2 + 3x^3 + \ldots $
It can be written as 
$ \Rightarrow S = (x + x^2 + x^3 + \ldots)+(x^2 + x^3 + \ldots)+(x^3 + \ldots)+\ldots $
$\Rightarrow S = (x + x^2 + x^3 + ...)+x(x + x^2 + ...)+x^2(x + ...) + \ldots $
$\Rightarrow S = ( 1+x+x^2+ .. )\times( x+x^2+.. )$
$\Rightarrow S = \frac{1}{1-x}\times\frac{x}{1-x}$
$\Rightarrow S = \frac{x}{(1-x)^2} $
Put $x =0.5$ you will get the answer 
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