Chapter 9.3, Problem 42ES

In his Treatise on the Configuration of Qualities and Motions (written in the 1350s), the French Bishop of Lisieux, Nicole Oresme, used a geometric method to find the sum of the series ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{k}{2^k}}=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdot \cdot \cdot$

In part $\left(a\right)$ of the accompanying figure, each term in the series is represented by the area of a rectangle, and in part $\left(b\right)$ the configuration in part $\left(a\right)$ has been divided into rectangles with areas $A_1,A_2,A_3,\mathrm{....}$ Find the sum $A_1+A_2+A_3+\cdot \cdot \cdot .$