<s>[INST] Summing $\sum^n_{k=1} \frac i {2^k}$ Could someone help me sum following series as $n$ goes to infinity:
$$\sum^n_{i=1} \frac i {2^i} = \frac 1{2^1} + \frac 2{2^2} + \frac 3{2^3} +\cdots+ \frac n{2^n}$$ 
I don't need the answer so much as being pointed towards what I need to know to figure out the answer...I assume there are different ways to do it. I used to know this stuff a long time ago but don't know where to begin now. Thanks...
 [/INST] Easier hint:
$$
\begin{array}{ccll}
\sum_1^\infty\frac{n}{x^n} &=& \frac{1}{x^1} + & \frac{1}{x^2} + &\frac{1}{x^3} + \ldots \\
&+& & \frac{1}{x^2} +& \frac{1}{x^3} + \ldots \\
&+& & & \frac{1}{x^3} + \ldots \\
&+& \cdots 
\end{array}
$$
Each of these is a simple geometric series (for example, row 2 sums to $\frac{1}{x^2} \frac{1}{1-1/x}$).
But each of those series is $1/x$ times the one above it.  So you have a second geometric sum to do. 
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