2026-04-03

Problem — 2026-04-03

Medium

Mediumalgebra

What is the value of $\displaystyle\sum_{n=1}^{\infty} \frac{n}{2^n}$ ?

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Let

S = \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \cdots

. Then

\frac{S}{2} = \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \cdots

. Subtracting:

S - \frac{S}{2} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 1

. So

\frac{S}{2} = 1

, giving

S = 2