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Solved: Convergence Write the remainder term Rn (x) for
Chapter 1, Problem 39RE(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.
\(f(x)=\ln (1+x),\ \ a=0,\ \ -\frac{1}{2}\ \leq\ x\ \leq\ \frac{1}{2}\)
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.
\(f(x)=\ln (1+x),\ \ a=0,\ \ -\frac{1}{2}\ \leq\ x\ \leq\ \frac{1}{2}\)
ANSWER:Solution 39RE
Step 1 of 2:
In this problem we need to find the remainder term in the taylor series expansion of f(x) = ln(1 +x) 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) =
= 0 , since
=
=
, then
=
= 1
=
)=
, then
=
= -1
=
=
, then
=
= 2
……………………. ……………(2)
Therefore , the Taylor series of with center 0 is as follows;
…….
ln(1+x) = x - +
-...........
=