How to determine this infinite sum

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To determine the result of the infinite sum (Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)) for x<1, the denominator simplifies to a geometric series, yielding 1/(1-x). The numerator can be derived using calculus, specifically by differentiating the denominator series, resulting in x/(1-x)^2. Consequently, the overall result of the series is x/(1-x). This provides a clear formula for evaluating the infinite sum in terms of x.
benf.stokes
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Hi,

How do i determine de result in terms of x of this series:

(Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)
For x<1

Thanks
 
Last edited:
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The denominator is a simple geometric series = 1/(1-x). The numerator series can be gotten using elementary calculus from the denominator series = xd/dx(1/(1-x))=x/(1-x)^2.
Net result = x/(1-x).
 

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