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Difference between revisions of "2006 SMT/Algebra Problems/Problem 10"

(Created page with "==Problem== Evaluate: <math> \sum_{k=1}^{\infty}\frac{k}{a^{k-1}} </math> for all <math> |a|<1 </math>. ==Solution== Let <math> S=\sum_{k=1}^{\infty}\frac{k}{a^{k-1}}=\frac{1}{a...")
 
(No difference)

Latest revision as of 10:35, 28 May 2012

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

See Also

2006 SMT/Algebra Problems

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