<s>[INST] Find the Sum of an Infinite Series Compute
$${1 \cdot \frac {1}{2} + 2 \cdot \frac {1}{4} + 3 \cdot \frac {1}{8} + \dots + n \cdot \frac {1}{2^n} + \dotsb.}$$
I tried creating a partition for this but no such luck.
What would the the equation to generate the sum for the nth term
 [/INST] Let
$$f(x)=\sum_{n=0}^{+\infty}\frac{x^n}{2^n}=\frac{2}{2-x}$$
with a radius of convergence $R=2$
$$f'(x)=\sum_{n=0}^{+\infty}n\frac{x^{n-1}}{2^n}=\frac{2}{(2-x)^2}$$
thus, your sum is $f'(1)=2$.
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