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)
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)