<s>[INST] Evaluate to find the sum of an infinite series $∑_{n=1}^\infty$ $n\over2^{n-1}$
or
1 + $2\over2$ + $3\over4$ + $4\over8$ + $5\over16$ + $\ldots$
How to go about evaluating the above, showing that it sums to 4?
 [/INST] $\sum_{n\geq 1} \frac{n}{2^{n-1}} = f'(1)$ where $f(x) = 2 \sum_{n\geq 0} \frac{x^n}{2^n} = 2 \frac{1}{1 - \frac{x}{2}}$ so that $f'(x) = 2 \frac{\frac{1}{2}}{\left(1-\frac{x}{2}\right)^2}$ and $f'(1) = 4$.
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