I get the formula:
ln(1 + x) = x - x²/2 + x³/3 - . . . for -1 < x ⩽⩽1.
How I prove it?
There are several ways. One is to calculate the Taylor's series of log(1+ x) about 0 (MacLaurin series).
Another is to use the fact that the geometric series $a+ at+ ar^2+ ar^3+ \cdot\cdot\cdot+ ar^n+ \cdot\cdot\cdot$ sums to $\frac{a}{1- r}$ for -1< r< 1. Take a= 1 and r= -x to get $\frac{1}{1+ x}= 1- x+ x^2- x^3+ \cdot\cdot\cdot+ (-1)^nx^n+ \cdot\cdot\cdot$. Then integrate term by term.