$$1+\frac{1}{3 \cdot 2^2}+ \frac{1}{5 \cdot 2^4}+ \frac{1}{7 \cdot 2^6}+ ....=$$
Attemp: As \begin{align*} 1+\frac1{3\cdot 2^2}+\frac1{5\cdot2^4}+\frac1{7\cdot 2^6}+\cdots&=\sum_{n\geqslant0}\frac1{(2n+1)2^{2n}}\\ &=2\sum_{n\geqslant0}\frac{(1/2)^{2n+1}}{2n+1}\\ \end{align*}
How do I do this log summation?