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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.

##### 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