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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

 


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