Hello, can you help me?
I have this problem: Calculate the sum of n/(2^n) from 1 to infinity, by the derivative of this serie: the sum of (x)^n from 1 to infinity.
I tried, doing the derivative but i can't find the exact relation between those series, doing it in wolfram, it is 2.

Question

Hello, can you help me?
I have this problem: Calculate the sum of n/(2^n) from 1 to infinity, by the derivative of this serie: the sum of (x)^n from 1 to infinity.
I tried, doing the derivative but i can't find the exact relation between those series, doing it in wolfram, it is 2.

holsteremission · Answer

If $|x|<1$, then$$f(x)=\sum_{n\ge0}x^n=\frac{1}{1-x}\implies f'(x)=\sum_{n\ge1}nx^{n-1}=-\frac{1}{(1-x)^2}$$

holsteremission · Answer

You'll get the value for your series by setting $x=\dfrac{1}{2}$.

hansgui · Answer

Thanks man (: