#Calculating infinite sum of (x * 0.51^x)

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subtle creek
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Putting arbitrary large values in my calculator, this converges, but I have no idea how to calculate it, I know 0.51^x is a geometric series and calculating it is simple but I have no idea what to do with the x term

raven tulip
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Let S = x * 0.51^x

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Look at 0.51 * S

subtle creek
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lemme give it a try

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0.51 * S = x * 0.51^(x+1)
0.51 * S + 0.51^(x+1) = (x+1) * 0.51^(x+1)
0.51 * S + 0.51^(x+1) = S
0.51^(x+1) = 0.49 * S
0.49 * S = 1/(1-0.51) = 1/0.49

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ok thanks, that did it