<s>[INST] Why $\sum_{k=1}^{\infty} \frac{k}{2^k} = 2$? 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 $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{2}$
 [/INST] Let $s = 1/2 + 2/4 + 3/8 + 4/16 + 5/32 + \dots$
Then $2s = 1 + 2/2 + 3/4 + 4/8 + 5/16 + \dots$
And then subtracting terms with similar denominators gives: $2s - s = 1  +1/2+1/4+1/8+1/16+\dots = 2$
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