## Proposition 8.4.15: Taylor Series for the Natural Log

Note that the series also converges forln(1+x) = x - 1/2 x^{2}+ 1/3 x^{3}- 1/4 x^{4}+ .... = for -1 < x 1

*x = 1*.

Instead of applying Taylor's theorem directly, we'll start with the geometric series and integrate both sides:

^{1}/_{1+x}dx =^{1}/_{1-(-x)}dx =

= (-x)^{n}dx = (-x)^{n}dx =

=^{(-1)n}/_{n+1}x^{n+1}=

where the computation is valid for *-1 < x < 1*.
It remains to prove the statement for *x = 1*. But this will
be a perfect application of Abel's Limit theorem:

converges for|x| < 1

converges as the alternating harmonic series

Therefore

ln(2) = ln(1+x) = =

according to Abel's Limit theorem.

### Fun Facts:

ln(2) = 1 - 1/2 + 1/3 - 1/4 + ...(but convergence is slow)

This fact has just been proven as an application of Abel's Limit theorem.

Alt. Harmonic Series approaching ln(2)=0.6931472