1. The problem is to find the value of the infinite series $$\sum_{n=1}^{\infty} \frac{n}{2^n}$$. 2. Recognize this as a weighted geometric series where the general term is $$\frac{n}{2^n}$$. 3. Recall the formula for the sum of the series $$\sum_{n=1}^{\infty} n x^n = \frac{x}{(1-x)^2}$$ for $$|x| < 1$$. 4. Here, $$x = \frac{1}{2}$$, which satisfies $$|x| < 1$$. 5. Substitute $$x = \frac{1}{2}$$ into the formula: $$\sum_{n=1}^{\infty} \frac{n}{2^n} = \frac{\frac{1}{2}}{(1 - \frac{1}{2})^2} = \frac{\frac{1}{2}}{\left(\frac{1}{2}\right)^2} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2.$$