AHSME 1962 P40

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Problem Statement

The limiting sum of the infinite series, $\frac{1}{10} + \frac{2}{10^2} + \frac{3}{10^3} + \dots$ whose $n$th term is $\frac{n}{10^n}$ is:

Problem Link

Solution

$$\text{Let } S \text{ be the limiting sum}$$ $$S = \dfrac{1}{10} + \dfrac{1}{100} + \dfrac{1}{1000} + \dots$$ $$ + \dfrac{1}{100} + \dfrac{1}{1000} + \dots$$ $$ + \dfrac{1}{1000} + \dfrac{1}{10000} + \dots$$ $$+\dots$$ $$S = \dfrac{\dfrac{1}{10}}{1 - \dfrac{1}{10}} + \dfrac{\dfrac{1}{100}}{1 - \dfrac{1}{10}} + \dots$$ $$S = \dfrac{1}{9} + \dfrac{1}{90} + \dfrac{1}{900} + \dots$$ $$S = \dfrac{\dfrac{1}{9}}{1 - \dfrac{1}{10}} = \dfrac{10}{81}$$