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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.

 Jun 28, 2020
 #1
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\(rT = \displaystyle\sum_{k=0}^\infty kr^{k + 1} = \sum_{k = 1}^\infty (k - 1)r^k = \sum_{k = 1}^\infty kr^k -\sum_{k = 1}^\infty r^k = T - S + 1\)

\(T = \dfrac{1 - S}{r - 1}\)

\(T = \dfrac{1 - \dfrac{1}{1 - r}}{r - 1} = \dfrac{r}{(1 - r)^2}\)

.
 
 Jun 28, 2020
edited by MaxWong  Jun 28, 2020
 #2
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Thank you. Have a nice day

 
Guest Jun 28, 2020

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