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Let , \(|r| < 1\) ,

                                                             \(S = \sum^{\infty}_{k=0}r^k\)

and 

 

                                                           \(T = \sum^{\infty}_{k=0}kr^k\)

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

 Jan 7, 2021
 #1
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Let f(r) = 1 + r + r^2 + ... = 1/(1 - r).

 

Taking the derivative: f'(r) = 1 + 2r + 3r^2 + ... = 2r/(1 - r)^2.

 

Then r + 2r^2 + 3r^3 + ... = 2r^2/(1 - r)^2.

 

Therefore, T = 2r^2/(1 - r)^2.

 Jan 7, 2021
 #2
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+1

Like so:

 

 Jan 7, 2021

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