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Solved: Remainder terms Find the remainder in the Taylor
Chapter 8, Problem 48E(choose chapter or problem)
Remainder terms Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim \limits_{n \rightarrow \infty} R_{n}(x)=0\) for all x in the interval of convergence.
f(x) = cos 2x, a = 0
Questions & Answers
QUESTION:
Remainder terms Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim \limits_{n \rightarrow \infty} R_{n}(x)=0\) for all x in the interval of convergence.
f(x) = cos 2x, a = 0
ANSWER: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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