 Solution

Subtract the Maclaurin series of $\log (1-x)$ from that of $\log (1+x)$. Their even powers cancel, givingThe full power series is $2{\sum }_{n\ge 0}{x}^{2n+1}/(2n+1)$, with radius of convergence one. For the real logarithm, the stated series applies on $|x|<1$.