Problem 48E

Remainder terms Find the remainder in the Taylor series centered at the point a for the following functions. Then show that for all x in the interval of convergence.

f(x)= cos 2x, a = 0

Solution 48E

Step 1:

First we find the Taylor series of f(x)=sin a ,at a=0 as follows

f(x)=cos 2x f(0)=1

f’(x)=-2 sin 2x f’(0)=0

f’’(x)=-4cos 2x f’’(0)=-4

f’’’(x)=8 sin 2x f’’’(0)=0

f’’’’(x)=16 cos 2x f’’’’(0)=16

Since the derivatives repeat in a cycle of four, we can write the Maclaurin series as follows:

=

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