Solution Found!
Convergence Write the remainder term Rn (x) for the Taylor
Chapter 1, Problem 37RE(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.
\(f(x)=e^{-x},\ \ a=0,\ \ -\infty\ <\ x\ <\ \infty\)
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.
\(f(x)=e^{-x},\ \ a=0,\ \ -\infty\ <\ x\ <\ \infty\)
ANSWER:Solution 37RE
Step 1 of 2:
In this problem we need to find the remainder term in the taylor series expansion of
at center a = 0 , and we have to prove that
= 0.
Thus the Taylor series of with center 0 is as follows;
Given : f(x) = , then f(0) =
= 1 , since
=
= -
, then
= -
= -1
=
=
, then
=
= 1
=
=
, then
=
= -1
…………………….and
…………(1)
Therefore , the Taylor series of with center 0 is as follows;
…….