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) = sin a, a = 0

Solution 47E

Step 1:

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

f(x)=sin x f(0)=0

f’(x)=cos x f’(0)=1

f’’(x)=-sin x f’’(0)=0

f’’’(x)=-cos x f’’’(0)=-1

f’’’’(x)=sin x f’’’’(0)=0

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

=

=

Step 2:

With , we have

Since is , we know that for all .

For ,