<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] 1/2 + 1/4 + 1/8 + 1/16 + ..... + 1/(2^n) + ..... = (1/2)*2 = 1
  1/4 + 1/8 + 1/16 + ..... + 1/(2^n) + ..... = [1/(2^2)]*2 = 1/2

        1/8 + 1/16 + ..... + 1/(2^n) + ..... = [1/(2^3)]*2 = 1/(2^2)

So on and so forth.
Hence, S = 1 + 1/2 + 1/(2^2) + ..... + 1/(2^n) + ..... = 1 + 1 = 2
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