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Power series for the square root

The square root can be expressed as a power series

 


Explanation

When calculating a power series for the square root you cannot work with , because the first derivative is not defined for x = 0. So we take

                     

as this will give a usefull result. We differentiate this function several times

                

             

               

That is an amazing regularity. We substitute this in the Taylor series

so that

In this way a calculator can determine the square root.

 


General format

You can write the power series as a sum

 


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