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Power series for the sine

The sine can be expressed as a power series

 


Explanation

As the sine can be differentiated continuously, you get







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







We substitute this in the Taylor series and find

so

 


Symmetry

The power series contains only odd numbers, so there is the symmetry

sin (−x) = −sin x

 


Example 1

You can see that sin (0) = 0, because

sin (0) = 0 − 0 + 0 − ... = 0

 


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