00 Show that 8 Σ anx" n=3 First, rewrite each series with the generic term x". 00 00 n+1 8 Σ anx" = 8 Σ n=3 n= _n-1 n+1 Σnb₂x n=4 = Σ 00 00 _n-1 + Σ nboxi = 4b4x³ + Σ [8an-1 + (n+1)bn+1]x². n=4 n = 4 n= How can the two series be combined into a single sum? O A. Adjust the sums so that the starting indices are the same. B. Adjust the sums so that the coefficients are the same. C. Adjust the sums so that the coefficients have the same index. D. Adjust the sums so that the summands are the same.
00 Show that 8 Σ anx" n=3 First, rewrite each series with the generic term x". 00 00 n+1 8 Σ anx" = 8 Σ n=3 n= _n-1 n+1 Σnb₂x n=4 = Σ 00 00 _n-1 + Σ nboxi = 4b4x³ + Σ [8an-1 + (n+1)bn+1]x². n=4 n = 4 n= How can the two series be combined into a single sum? O A. Adjust the sums so that the starting indices are the same. B. Adjust the sums so that the coefficients are the same. C. Adjust the sums so that the coefficients have the same index. D. Adjust the sums so that the summands are the same.
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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![**Explanation of the Mathematical Problem**
The problem begins with a mathematical expression:
\[
8 \sum_{n=3}^{\infty} a_n x^{n+1} + \sum_{n=4}^{\infty} nb_n x^{n-1} = 4b_4x^3 + \sum_{n=4}^{\infty} [8a_{n-1} + (n+1)b_n + 1] x^n.
\]
**Objective:**
To manipulate and combine the series into a single sum by rewriting them with a generic term \( x^n \).
**Steps Involved:**
1. **Rewrite Each Series Using \( x^n \):**
- The first series, \( 8 \sum_{n=3}^{\infty} a_n x^{n+1} \), is rewritten with a common factor of 8, changing the exponent and starting index:
\[
8 \sum_{n=3}^{\infty} a_n x^{n+1} = 8 \sum_{n=\square}^{\infty} [ \square ] x^n
\]
- The second series, \( \sum_{n=4}^{\infty} nb_n x^{n-1} \), is similarly rewritten:
\[
\sum_{n=4}^{\infty} nb_n x^{n-1} = \sum_{n=\square}^{\infty} [ \square ] x^n
\]
2. **Combining into a Single Sum:**
- The question then asks: "How can the two series be combined into a single sum?"
- Here are the options provided:
- **A.** Adjust the sums so that the starting indices are the same.
- **B.** Adjust the sums so that the coefficients are the same.
- **C.** Adjust the sums so that the coefficients have the same index.
- **D.** Adjust the sums so that the summands are the same.
**Conclusion:**
To combine the series, the correct choice involves making adjustments so that terms can be summed together cohesively, considering factors like starting indices, coefficients, indices, and summands, based on the given options. The restructuring allows the operation of a singular, unified series representation.
This helps in understanding the process of transforming and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa827acfe-a0bc-46c0-ab61-62657df3b5db%2F9f0d06a9-ad30-44cb-956f-2aa0b2338e26%2F97bk75_processed.png&w=3840&q=75)
Transcribed Image Text:**Explanation of the Mathematical Problem**
The problem begins with a mathematical expression:
\[
8 \sum_{n=3}^{\infty} a_n x^{n+1} + \sum_{n=4}^{\infty} nb_n x^{n-1} = 4b_4x^3 + \sum_{n=4}^{\infty} [8a_{n-1} + (n+1)b_n + 1] x^n.
\]
**Objective:**
To manipulate and combine the series into a single sum by rewriting them with a generic term \( x^n \).
**Steps Involved:**
1. **Rewrite Each Series Using \( x^n \):**
- The first series, \( 8 \sum_{n=3}^{\infty} a_n x^{n+1} \), is rewritten with a common factor of 8, changing the exponent and starting index:
\[
8 \sum_{n=3}^{\infty} a_n x^{n+1} = 8 \sum_{n=\square}^{\infty} [ \square ] x^n
\]
- The second series, \( \sum_{n=4}^{\infty} nb_n x^{n-1} \), is similarly rewritten:
\[
\sum_{n=4}^{\infty} nb_n x^{n-1} = \sum_{n=\square}^{\infty} [ \square ] x^n
\]
2. **Combining into a Single Sum:**
- The question then asks: "How can the two series be combined into a single sum?"
- Here are the options provided:
- **A.** Adjust the sums so that the starting indices are the same.
- **B.** Adjust the sums so that the coefficients are the same.
- **C.** Adjust the sums so that the coefficients have the same index.
- **D.** Adjust the sums so that the summands are the same.
**Conclusion:**
To combine the series, the correct choice involves making adjustments so that terms can be summed together cohesively, considering factors like starting indices, coefficients, indices, and summands, based on the given options. The restructuring allows the operation of a singular, unified series representation.
This helps in understanding the process of transforming and
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