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Power series for the cosine

The cosine can be expressed as a power series

 


Explanation

As the cosine can be differentiated continuously, you get







The point x = 0 gives cos (0) = 1 and sin (0) = 0 so







We substitute this in the Taylor series and find

so

 


Symmetry

The formula only contains even numbers, so there is the symmetry

 


Example 1

You can see that cos (0) = 1, because

 


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