Power series for the logarithm
The natural logarithm can be expressed as a power series
Explanation
In the sum of the geometrical series
we substitute r = – t and write
Integration gives
Calculation of this inversed derivative of the logarithm shows
Substitution of x = x – 1 gives
Example 1
You can see that ln 1 = 0, because
ln 1 = (0) − (0) + (0) − (0) + ⋯ = 0