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?

For these sorts of problems it is helpful to start with geometric series: $$\sum_{n=0}^\infty x^n = \frac{1}{1-x}.$$ If we differentiate this series we find $$\sum_{n=0}^\infty nx^{n-1} = \sum_{n=1}^\infty nx^{n-1}  = \frac{1}{(1-x)^2}.$$ This series is valid for all $x \in (-1,1)$.
