Let |r|<1, \(S = \sum_{k=0}^{\infty} r^k\) and \(T = \sum_{k=0}^{\infty} k r^k\) Our approach is to write T as a geometric series in terms of S and r. Give a closed form expression for T in terms of r.
\(rT = \displaystyle\sum_{k=0}^\infty kr^{k + 1} = \sum_{k = 1}^\infty (k - 1)r^k = \sum_{k = 1}^\infty kr^k -\sum_{k = 1}^\infty r^k = T - S + 1\)
\(T = \dfrac{1 - S}{r - 1}\)
\(T = \dfrac{1 - \dfrac{1}{1 - r}}{r - 1} = \dfrac{r}{(1 - r)^2}\)
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