Proposition 8.4.15: Taylor Series for the Natural Log
ln(1+x) = x - 1/2 x2 + 1/3 x3 - 1/4 x4 + .... = for -1 < x 1Note that the series also converges for 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 xn+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
ln(2) = ln(1+x) = =
according to Abel's Limit theorem.
- 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