1) Suppose we know that ƒ(¹)(4) = = (-1)^n! 3n(n+1) and the Taylor series of f centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates ƒ(5) with error less than 0.0002.
1) Suppose we know that ƒ(¹)(4) = = (-1)^n! 3n(n+1) and the Taylor series of f centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates ƒ(5) with error less than 0.0002.
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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![### Problem Statement
1) Suppose we know that
\[f^{(n)}(4) = \frac{(-1)^n n!}{3^{n(n+1)}}\]
and the Taylor series of \(f\) centered at 4 converges to \(f(x)\) for all \(x\) in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates \(f(5)\) with error less than 0.0002.
### Steps to Solve
To approach the problem, follow these steps:
1. **Understand the Taylor Series**:
The Taylor series of \(f\) centered at \(x = 4\) is given by:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(4)}{n!} (x-4)^n \]
2. **Fifth-degree Taylor Polynomial**:
The fifth-degree Taylor polynomial \(P_5(x)\) centered at \(x=4\) is:
\[ P_5(x) = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} (x-4)^n \]
3. **Calculate \(f^{(n)}(4)\)**:
Given the formula for \(f^{(n)}(4)\):
\[ f^{(n)}(4) = \frac{(-1)^n n!}{3^{n(n+1)}} \]
4. **Evaluate Polynomial at \(x = 5\)**:
Substitute \(x = 5\) into \(P_5(x)\):
\[ P_5(5) = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} (5-4)^n = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} \]
5. **Bound the Error**:
The error term \(R_5(x)\) for the Taylor polynomial approximation is given by:
\[ R_5(x) = \frac{f^{(6)}(c)}{6!} (x-4)^6 \]
for some \(c\) between 4 and 5. Ensure that this error \(R_5](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3da0cd1-4491-4699-a531-553fdaf9ab08%2Ff07fe574-b236-4cf4-936a-a0b5209172a0%2F5pxspl_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
1) Suppose we know that
\[f^{(n)}(4) = \frac{(-1)^n n!}{3^{n(n+1)}}\]
and the Taylor series of \(f\) centered at 4 converges to \(f(x)\) for all \(x\) in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates \(f(5)\) with error less than 0.0002.
### Steps to Solve
To approach the problem, follow these steps:
1. **Understand the Taylor Series**:
The Taylor series of \(f\) centered at \(x = 4\) is given by:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(4)}{n!} (x-4)^n \]
2. **Fifth-degree Taylor Polynomial**:
The fifth-degree Taylor polynomial \(P_5(x)\) centered at \(x=4\) is:
\[ P_5(x) = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} (x-4)^n \]
3. **Calculate \(f^{(n)}(4)\)**:
Given the formula for \(f^{(n)}(4)\):
\[ f^{(n)}(4) = \frac{(-1)^n n!}{3^{n(n+1)}} \]
4. **Evaluate Polynomial at \(x = 5\)**:
Substitute \(x = 5\) into \(P_5(x)\):
\[ P_5(5) = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} (5-4)^n = \sum_{n=0}^{5} \frac{f^{(n)}(4)}{n!} \]
5. **Bound the Error**:
The error term \(R_5(x)\) for the Taylor polynomial approximation is given by:
\[ R_5(x) = \frac{f^{(6)}(c)}{6!} (x-4)^6 \]
for some \(c\) between 4 and 5. Ensure that this error \(R_5
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