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 $1 + 2 + 3 + ... + n = \dfrac{n(n+1)}{2}$

Since infinite series with nonnegative terms can be rearranged arbitrarily,
$$\sum_{i=1}^\infty \frac{i}{2^i} =
\sum_{i=1}^\infty \sum_{j=1}^i \frac{1}{2^i}
= \sum_{j=1}^\infty \sum_{i=j}^\infty \frac{1}{2^i}
= \sum_{j=1}^\infty \frac{1}{2^{j-1}}
= 2
$$
More graphically,
  1/2 + 2/4 + 3/8 + 4/16 + ...

= 1/2 + 1/4 + 1/8 + 1/16 + ...    (= 1)
      + 1/4 + 1/8 + 1/16 + ...    (= 1/2)           
            + 1/8 + 1/16 + ...    (= 1/4)
                  + 1/16 + ...    (= 1/8)
                        ....       ....

