15a) n=0 (-1)² 7³² (2n)!9" 2n+1 15b) (−1)" 3+¹ 00 n=0n!5"- 15c) √ x³ sin(x²)dx
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
Use power series to evaluate the following:
![### Advanced Mathematics Problems: Series and Integral
#### Problem 15a:
\[ \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{(2n)! 9^n} \]
#### Problem 15b:
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 3^{n+1}}{n! \, 5^{n-1}} \]
#### Problem 15c:
\[ \int x^3 \sin(x^7) \, dx \]
### Explanation
These problems involve advanced mathematical operations:
1. **Problem 15a** and **15b**: These are infinite series that involve summing terms of a sequence.
- In **15a**, the series has a general term involving powers of π, factorial terms, and powers of 9.
- In **15b**, the series has a general term involving powers of 3, factorial terms, and powers of 5.
2. **Problem 15c**: This is an indefinite integral involving a product of a polynomial function \(x^3\) and a sine function with \(x^7\) as the argument.
These problems illustrate the complexity and beauty of infinite series and integrals in higher mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5456dffd-fe15-479d-aec2-2e26d1f45f3e%2F9883e21d-6dec-4784-b0b5-213297ba3a7b%2Fndcwy8_processed.png&w=3840&q=75)
Transcribed Image Text:### Advanced Mathematics Problems: Series and Integral
#### Problem 15a:
\[ \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{(2n)! 9^n} \]
#### Problem 15b:
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 3^{n+1}}{n! \, 5^{n-1}} \]
#### Problem 15c:
\[ \int x^3 \sin(x^7) \, dx \]
### Explanation
These problems involve advanced mathematical operations:
1. **Problem 15a** and **15b**: These are infinite series that involve summing terms of a sequence.
- In **15a**, the series has a general term involving powers of π, factorial terms, and powers of 9.
- In **15b**, the series has a general term involving powers of 3, factorial terms, and powers of 5.
2. **Problem 15c**: This is an indefinite integral involving a product of a polynomial function \(x^3\) and a sine function with \(x^7\) as the argument.
These problems illustrate the complexity and beauty of infinite series and integrals in higher mathematics.
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