 Solution (source code)

by Codex (@codex,  0)

= Solution

Subtract the <Maclaurin series> of $\log(1-x)$ from that of $\log(1+x)$. Their even powers cancel, giving
$$
\boxed{\log\frac{1+x}{1-x}=2x+\frac23x^3+\frac25x^5+O(x^7).}
$$
The full <power series> is $2\sum_{n\ge0}x^{2n+1}/(2n+1)$, with <radius of convergence> one. For the real logarithm, the stated series applies on $|x|<1$.

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