Games with Logarithms

It is well-known that
$$
\log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-+\ \cdots\ \text{for}\ 0\le x\le1,
$$
where the $-+\ \cdots$ notation is used to indicate that successive terms are
alternately subtracted and added.