Let f(x) = ln(1+x). Below are some derivatives of f(x) which you can use to answer parts [a-d] f'(x) = 1+x f"(x) = - | f"'(x) = a 2 (1+x)3 f" (x) = -a2) 6. (1+x)4 fn+1(x) = (1+z)n+Í [a] Find the third Taylor polynomial, T3(x), around a = 0 for f(x). [b] Approximate In(1.1) using T3(x) around a = 0. The answer can be left as a sum of fractions. [c] Find a bound for the error in this approximation.
Let f(x) = ln(1+x). Below are some derivatives of f(x) which you can use to answer parts [a-d] f'(x) = 1+x f"(x) = - | f"'(x) = a 2 (1+x)3 f" (x) = -a2) 6. (1+x)4 fn+1(x) = (1+z)n+Í [a] Find the third Taylor polynomial, T3(x), around a = 0 for f(x). [b] Approximate In(1.1) using T3(x) around a = 0. The answer can be left as a sum of fractions. [c] Find a bound for the error in this approximation.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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COURSE: Mathematical/
TOPIC: Taylor Series
![**Taylor Series and Approximation of \( \ln(1 + x) \)**
Let \( f(x) = \ln(1 + x) \). Below are some derivatives of \( f(x) \) which you can use to answer parts [a-d]:
\[
f'(x) = \frac{1}{1+x}
\]
\[
f''(x) = -\frac{1}{(1+x)^2}
\]
\[
f'''(x) = \frac{2}{(1+x)^3}
\]
\[
f''''(x) = -\frac{6}{(1+x)^4}
\]
\[
f^{n+1}(x) = \frac{(-1)^n (n!)}{(1+x)^{n+1}}
\]
**[a]** Find the third Taylor polynomial, \( T_3(x) \), around \( a = 0 \) for \( f(x) \).
**[b]** Approximate \( \ln(1.1) \) using \( T_3(x) \) around \( a = 0 \). The answer can be left as a sum of fractions.
**[c]** Find a bound for the error in this approximation.
**[d]** Prove that the Taylor series for \( f(x) \), which is
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n+1} (x)^n}{n}
\]
(you don’t need to find this formula), converges to \( \ln(1 + x) \) for all \( 0 < x < 1 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaa9c753-6b68-41b4-8837-61aa8aa0a19b%2F53e70742-c996-4494-a6ac-daf329a0dbd3%2Fy1rrqr_processed.png&w=3840&q=75)
Transcribed Image Text:**Taylor Series and Approximation of \( \ln(1 + x) \)**
Let \( f(x) = \ln(1 + x) \). Below are some derivatives of \( f(x) \) which you can use to answer parts [a-d]:
\[
f'(x) = \frac{1}{1+x}
\]
\[
f''(x) = -\frac{1}{(1+x)^2}
\]
\[
f'''(x) = \frac{2}{(1+x)^3}
\]
\[
f''''(x) = -\frac{6}{(1+x)^4}
\]
\[
f^{n+1}(x) = \frac{(-1)^n (n!)}{(1+x)^{n+1}}
\]
**[a]** Find the third Taylor polynomial, \( T_3(x) \), around \( a = 0 \) for \( f(x) \).
**[b]** Approximate \( \ln(1.1) \) using \( T_3(x) \) around \( a = 0 \). The answer can be left as a sum of fractions.
**[c]** Find a bound for the error in this approximation.
**[d]** Prove that the Taylor series for \( f(x) \), which is
\[
\sum_{n=1}^{\infty} \frac{(-1)^{n+1} (x)^n}{n}
\]
(you don’t need to find this formula), converges to \( \ln(1 + x) \) for all \( 0 < x < 1 \).
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