What is the sum of the series for (0.25)^n/n
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What is the sum of the series for (0.25)^n/n

[From: ] [author: ] [Date: 12-04-29] [Hit: ]
-ln(1 - x) = x + x^2/2 + x^3/3 + ............
from n = 1 to inf

-
Use the Taylor series for ln(x) centered at x=1:

ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ...

That's nearly your series for x=0.25, except for the alternating signs. "Un-alternate" them by replacing x with -x:

ln(1 - x) = -x - x^2/2 - x^3/3 - x^4/4 - ...
-ln(1 - x) = x + x^2/2 + x^3/3 + ...

...so your series sum is -ln(1 - 1/4) = -ln(3/4) = ln(4/3)
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