The infinite series $$\frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots + \frac{n}{2^n} + \cdots$$is​

Correct option is C

Solution:

​$$\begin{aligned}&\text{Given series: } S = \sum_{n=1}^{\infty} \frac{n}{2^n} \\&\text{We use the identity: } \sum_{n=1}^{\infty} \frac{n}{r^n} = \frac{r}{(r - 1)^2}, \quad \text{for } r > 1 \\&\text{Here, } r = 2 \Rightarrow \sum_{n=1}^{\infty} \frac{n}{2^n} = \frac{2}{(2 - 1)^2} = \frac{2}{1} = 2 \\&\therefore \text{The series is convergent and its sum is } \boxed{2}\end{aligned}$$

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