1. Find the first three nonzero terms for the Taylor series of f(x) = x-² about a = -1. A. 1+2x+3x² OB. 1-2x+3x² OC. 1+ 2(x+1)+ 3(x + 1)² OD. 1-2(x+1)+3(x + 1)² OE. None of these
1. Find the first three nonzero terms for the Taylor series of f(x) = x-² about a = -1. A. 1+2x+3x² OB. 1-2x+3x² OC. 1+ 2(x+1)+ 3(x + 1)² OD. 1-2(x+1)+3(x + 1)² OE. None of these
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. Find the first three nonzero terms for the Taylor series of \( f(x) = x^{-2} \) about \( a = -1 \).
**Options:**
- **A.** \( 1 + 2x + 3x^2 \)
- **B.** \( 1 - 2x + 3x^2 \)
- **C.** \( 1 + 2(x + 1) + 3(x + 1)^2 \)
- **D.** \( 1 - 2(x + 1) + 3(x + 1)^2 \)
- **E.** None of these
**User Response:**
- **Entered:** B
- **Answer Preview:** B
**Feedback:**
- The answer above is NOT correct.
**Explanation:**
The Taylor series expansion involves finding derivatives of \( f(x) \) evaluated at \( a = -1 \) and using the formula for the Taylor series:
\[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]
Carefully perform the calculations to derive the correct first three nonzero terms for the function series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2Fff663660-fd18-4f21-b7e9-c5a6ca2d95fb%2F6jpkvo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. Find the first three nonzero terms for the Taylor series of \( f(x) = x^{-2} \) about \( a = -1 \).
**Options:**
- **A.** \( 1 + 2x + 3x^2 \)
- **B.** \( 1 - 2x + 3x^2 \)
- **C.** \( 1 + 2(x + 1) + 3(x + 1)^2 \)
- **D.** \( 1 - 2(x + 1) + 3(x + 1)^2 \)
- **E.** None of these
**User Response:**
- **Entered:** B
- **Answer Preview:** B
**Feedback:**
- The answer above is NOT correct.
**Explanation:**
The Taylor series expansion involves finding derivatives of \( f(x) \) evaluated at \( a = -1 \) and using the formula for the Taylor series:
\[ f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]
Carefully perform the calculations to derive the correct first three nonzero terms for the function series.
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