TITLE: Summation of this series ${1\over 2}\left[1+{1 \over 3}\left({1 \over 4}\right)^2+ {1 \over 5}\left({1 \over 4}\right)^4+\cdots\right]$
QUESTION [7 upvotes]: I have this series,
$${1\over 2}\left[1+{1 \over 3}\left({1 \over 4}\right)^2+ {1 \over 5}\left({1 \over 4}\right)^4+\cdots\right]$$
I have shown that this series converges. 
It looks like some combination of geometric series and harmonic series to me. 
I am not able to proceed from here. I would love to have some hint. 
Sorry this time I have nothing to write in "my efforts" as I am completely stuck.

REPLY [2 votes]: Hint:You may use following function:
$$S=1+\frac{1}{3}x^2+\frac{1}{5}x^4+ . . . $$
Multiplying both sides by x we get:
$$S.x=x+\frac{1}{3}x^3+\frac{1}{5}x^5+ . . .$$
Taking derivative we get:
$$ S'x+ S=1+x^2+x^4 +. . . (x^2)^n=\frac{(x^2)^{n+1} -1}{x^2-1}$$
Taking integral we get:
$$S.x=\int \frac{(x^2)^{n+1} -1}{x^2-1}$$
$$S=\frac{1}{x}\int \frac{(x^2)^{n+1} -1}{x^2-1}$$
Put $x=\frac {1}{4}$.