∞ Е 5пспхп-1 + E 20пхп n=1 n = 0 + 1
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
Proceed as in Example 3 in Section 6.1 to rewrite the given expression using a single power series whose general term involves
![The image contains two mathematical summations. Below is the transcription and explanation:
**Mathematical Expression:**
\[ \sum_{n=1}^{\infty} 5nC_n x^{n-1} + \sum_{n=0}^{\infty} 2C_n x^{n+1} \]
**Explanation:**
1. **First Summation (\(\sum_{n=1}^{\infty} 5nC_n x^{n-1}\))**: This summation starts at \(n=1\) and continues to infinity. The term inside the summation is \(5nC_n x^{n-1}\), where:
- \(5\) is a constant coefficient.
- \(n\) represents the current term index.
- \(C_n\) represents a sequence of coefficients.
- \(x^{n-1}\) represents the variable \(x\) raised to the power of \(n-1\).
2. **Second Summation (\(\sum_{n=0}^{\infty} 2C_n x^{n+1}\))**: This summation starts at \(n=0\) and continues to infinity. The term inside the summation is \(2C_n x^{n+1}\), where:
- \(2\) is a constant coefficient.
- \(C_n\) represents a sequence of coefficients.
- \(x^{n+1}\) represents the variable \(x\) raised to the power of \(n+1\).
These summations are typically used in mathematical analysis, particularly in series expansion problems, and can be found in calculus and differential equations. The coefficients \(C_n\) often depend on the context of the problem being solved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc59b3e01-f4fe-4ed0-a002-b05c91db4102%2F8e56ec47-6b3c-44d2-8f47-0cb953133029%2F21btg7h_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains two mathematical summations. Below is the transcription and explanation:
**Mathematical Expression:**
\[ \sum_{n=1}^{\infty} 5nC_n x^{n-1} + \sum_{n=0}^{\infty} 2C_n x^{n+1} \]
**Explanation:**
1. **First Summation (\(\sum_{n=1}^{\infty} 5nC_n x^{n-1}\))**: This summation starts at \(n=1\) and continues to infinity. The term inside the summation is \(5nC_n x^{n-1}\), where:
- \(5\) is a constant coefficient.
- \(n\) represents the current term index.
- \(C_n\) represents a sequence of coefficients.
- \(x^{n-1}\) represents the variable \(x\) raised to the power of \(n-1\).
2. **Second Summation (\(\sum_{n=0}^{\infty} 2C_n x^{n+1}\))**: This summation starts at \(n=0\) and continues to infinity. The term inside the summation is \(2C_n x^{n+1}\), where:
- \(2\) is a constant coefficient.
- \(C_n\) represents a sequence of coefficients.
- \(x^{n+1}\) represents the variable \(x\) raised to the power of \(n+1\).
These summations are typically used in mathematical analysis, particularly in series expansion problems, and can be found in calculus and differential equations. The coefficients \(C_n\) often depend on the context of the problem being solved.
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