2006 SMT/Algebra Problems/Problem 10

Problem

Evaluate: $\sum_{k=1}^{\infty}\frac{k}{a^{k-1}}$ for all $|a|<1$ .

Solution

Let $S=\sum_{k=1}^{\infty}\frac{k}{a^{k-1}}=\frac{1}{a^{0}}+\frac{2}{a^{1}}+\frac{3}{a^{2}}+\frac{4}{a^{3}}+\cdots$ .

We have $\frac{S}{a}=\frac{1}{a^{1}}+\frac{2}{a^{2}}+\frac{3}{a^{3}}+\cdots$ .

Subtracting these two equations, we get $S-\frac{S}{a}=\frac{1}{a^{0}}+\frac{1}{a^{1}}+\frac{1}{a^{2}}+\frac{1}{a^{3}}+\cdots=\frac{\frac{1}{a^{0}}}{1-\frac{1}{a}}=\frac{a}{a-1}$ .

Therefore, $S=\frac{\frac{a}{a-1}}{1-\frac{1}{a}}=\frac{\frac{a^2}{a-1}}{a-1}=\boxed{\frac{a^2}{a^2-2a+1}}$ .

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2006 SMT/Algebra Problems/Problem 10

Problem

Solution

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