Maeckes logo

<    1      2    >

Power series for the square root

The square root can be expressed as a power series



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