Power series for the logarithm
The logarithm can be expressed as a power series
Explanation
When calculating a power series for the logarithm you cannot work with f (x) = ln x, because that is not defined for x = 0. So we take
as this will give a usefull result. We differentiate this function several times
what can be written as
That is an amazing regularity. We substitute this in the Taylor series
Substitution of x = x – 1 gives the final form
Example 1
You can see that ln 1 = 0, because
ln 1 = (0) − (0) + (0) − (0) + ⋯ = 0
Example 2
You can see that ln 2 is the conditionally convergent alternating harmonic series
ln 2 = 1 − ½ + ⅓ − ¼ + ⋯