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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