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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.

 Mar 27, 2021
 #1
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T = r/(1 - r)^3.

 Mar 27, 2021

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