How can we show that $ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $?

Question

How can we prove the following?

$$ \sum_{n=1}^{\infty} \frac{n}{2^n} = 2 $$

It would be great to see multiple ways, or hints, about how this can be proven.

I know this is a power series $\sum_{n=0}^\infty c_n \, (x-a)^n$ where $c_n = n$ and $x = \frac{1}{2}$ and $a = 0$, but otherwise I am stuck.

Answer 1

Consider $f(x)=\sum\limits_{i=0}^\infty x^i=\frac{1}{1-x}$ for $|x|<1$. Then

$$f'(x)=\sum\limits_{i=0}^\infty i x^{i-1}=\sum\limits_{i=1}^\infty i x^{i-1}=\frac1{(1-x)^2}$$ for $|x|<1$. Plugging in $x=\frac12$, we have

$$\frac12f'(1/2)=\sum_{i=1}^\infty\frac i{2^i}=\frac1{2 \left(1-\frac12\right)^2}=2$$

So $$\sum\limits_{i=1}^\infty\frac i{2^i}=2$$

Answer 2

Let $$ I=\sum_{i=1}^\infty\frac{i}{2^i}. $$ Then $$ 2I=\sum_{i=1}^\infty\frac{i}{2^{i-1}}=1+\sum_{i=2}^\infty\frac{i}{2^{i-1}} =1+\sum_{i=1}^\infty\frac{i+1}{2^{i}}=1+I+\sum_{n=1}^\infty\frac{1}{2^i}=2+I. $$ Thus $I=2$.

Note. If $2$ in $\frac{i}{2^i}$ is replaced by $a>1$, then $I$ becomes $\frac{2}{a-1}$.

Answer 3

As a change of pace from the usual derivative, we can show that the result is an immediate consequence of the summagration-by-parts formula for two sequences:

$$\sum_{k=m}^{n}f_{k}\left(g_{k+1}-g_{k}\right)=\left[f_{n+1}g_{n+1}-f_{m}g_{m}\right]-\sum_{k=m}^{n}g_{k+1}\left(f_{k+1}-f_{k}\right).$$

Letting $f_{k}=k$ and $g_{k}=-2^{1-k}$, we have:

$$\sum_{k=m}^{n}k\,2^{-k}=\left[-(n+1)2^{1-n-1}+m\,2^{1-m}\right]+\sum_{k=m}^{n}2^{1-k-1}\left(k+1-k\right).$$

It should be fairly clear where to go from there.