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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 1
Note 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

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
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