Maeckes logo

<    1      2    >


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.

 


Example 1

Substitution of x = 1 in the power series gives 2 as

and that produces the value 1.414213562… that is shown on your calculator.

 


General format

You can write the power series as a sum

 


Deutsch   Español   Français   Nederlands   中文   Русский