he origin for the function 5+2 sin(x²). : -1 | |-100 | 13 : -2 = 2
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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Question
![**Part 1 of 4**
Find the coefficient of \(x^6\) in the Taylor series expansion centered at the origin for the function
\[ f(x) = 6 + 2 \sin(x^2). \]
1. Coefficient of \(x^6 = -1\)
2. Coefficient of \(x^6 = -2\)
3. Coefficient of \(x^6 = \frac{1}{3}\)
4. Coefficient of \(x^6 = -\frac{1}{3}\)
5. Coefficient of \(x^6 = 2\)
---
**Part 2 of 4**
Find a power series representation for the function
\[ f(x) = x \sin(2x) \]
on \((- \infty, \infty)\).
1. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{2n}}{2n-1} x^{2n} \)
2. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{n-1}}{n-1} x^{n} \)
3. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{2n-1}}{(2n-1)!} x^{2n} \)
4. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^n}{(n-1)!} x^{n} \)
5. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{(n-1)!} x^{n} \)
6. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{(2n)!} x^{2n} \)
---
**Part 3 of 4**
Find the Taylor series centered at the origin for the function
\[ f(x) = x \cos(2x). \]
1. \( f(x) = \sum_{n=0}^{\infty}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc88a61a8-930a-4496-81c3-aab24a173799%2Fa6023342-bc8e-429c-9247-7e14d5565392%2Fy8gppbi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Part 1 of 4**
Find the coefficient of \(x^6\) in the Taylor series expansion centered at the origin for the function
\[ f(x) = 6 + 2 \sin(x^2). \]
1. Coefficient of \(x^6 = -1\)
2. Coefficient of \(x^6 = -2\)
3. Coefficient of \(x^6 = \frac{1}{3}\)
4. Coefficient of \(x^6 = -\frac{1}{3}\)
5. Coefficient of \(x^6 = 2\)
---
**Part 2 of 4**
Find a power series representation for the function
\[ f(x) = x \sin(2x) \]
on \((- \infty, \infty)\).
1. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{2n}}{2n-1} x^{2n} \)
2. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{n-1}}{n-1} x^{n} \)
3. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^{2n-1}}{(2n-1)!} x^{2n} \)
4. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{2^n}{(n-1)!} x^{n} \)
5. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{(n-1)!} x^{n} \)
6. \( f(x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{(2n)!} x^{2n} \)
---
**Part 3 of 4**
Find the Taylor series centered at the origin for the function
\[ f(x) = x \cos(2x). \]
1. \( f(x) = \sum_{n=0}^{\infty}
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