TITLE: Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$?
QUESTION [35 upvotes]: Can you please explain why
$$
\sum_{k=1}^{\infty} \dfrac{k}{2^k} =
\dfrac{1}{2} +\dfrac{ 2}{4} + \dfrac{3}{8}+ \dfrac{4}{16} +\dfrac{5}{32} + \dots =
2
$$
I know $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{2}$

REPLY [18 votes]: Here is a visual proof that $$\frac14+\frac28+\frac3{16}+\frac{4}{32}+\frac{5}{64}+\cdots=1$$ which is the same relation after dividing by $2$ on each side:

If it is not clear how, here some areas are labeled (although I prefer the aesthetics of the unlabeled version). The rectangles fill up this $1\times1$ square, and for $i\in\mathbb{N}$, there are $i$ rectangles of area $\frac{1}{2^{n+1}}$.