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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

 


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