Solution Found!
Answer: Convergence Write the remainder term Rn (x) for
Chapter 1, Problem 40RE(choose chapter or problem)
Write the remainder term \(R_n(x)\) for the Taylor series for the following functions centered at the given point a. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all x in the given interval.
\(f(x)=\sqrt{1+x},\ \ a=0,\ \ -\frac{1}{2}\ \leq\ x\ \leq\ \frac{1}{2}\)
Questions & Answers
QUESTION:
Write the remainder term \(R_n(x)\) for the Taylor series for the following functions centered at the given point a. Then show that \(\lim _{n \rightarrow \infty} R_{n}(x)=0\) for all x in the given interval.
\(f(x)=\sqrt{1+x},\ \ a=0,\ \ -\frac{1}{2}\ \leq\ x\ \leq\ \frac{1}{2}\)
ANSWER:Solution 40RE
Step 1:
First we find the Taylor series of ,at a=0 as follows
Thus the Taylor series of with center 0 is as follows