134) Σ 1 * +1 ₂² n=1 ne"
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
Determine if the following series is convergent of divergent. Must state methods or theorems used.
![### Infinite Series Expressions
#### 13d)
\[
\sum_{n=1}^{\infty} \frac{1}{ne^{n^2+1}}
\]
This mathematical expression represents an infinite series where the general term is given by \(\frac{1}{n e^{n^2 + 1}}\). The summation starts from \(n=1\) and extends to infinity.
#### 13e)
\[
\sum_{n=1}^{\infty} \frac{(-1)^n 3^{n+2}}{(n+1)!}
\]
This infinite series involves a term that includes an alternating sign denoted by \((-1)^n\), which causes the terms to alternate between positive and negative. The general term is \(\frac{(-1)^n 3^{n+2}}{(n+1)!}\), and the series summation begins from \(n=1\) and continues to infinity.
#### 13f)
\[
\sum_{n=1}^{\infty} \left( \frac{3n^2 + n + 4}{2n^2 + 1} \right)^n
\]
The third expression represents an infinite series with the general term \(\left( \frac{3n^2 + n + 4}{2n^2 + 1} \right)^n\). This series also starts from \(n=1\) and extends indefinitely.
### Detailed Explanation
- **Infinite Series**: An infinite series is the sum of the terms of an infinite sequence. The expressions shown above involve different types of infinite series, featuring variable components in the terms.
- **Alternating Series**: Some series alternate in sign (e.g., 13e), indicated by the \((-1)^n\) factor, which means the series adds and subtracts terms successively.
These series are often studied in mathematical analysis and have applications in various fields such as physics, engineering, and finance. Analyzing the convergence or divergence of these series is a fundamental aspect of understanding their behavior and application.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5456dffd-fe15-479d-aec2-2e26d1f45f3e%2Ff53a759e-7653-4c84-bc52-5eeec4a933fd%2Fts3h5zv_processed.png&w=3840&q=75)
Transcribed Image Text:### Infinite Series Expressions
#### 13d)
\[
\sum_{n=1}^{\infty} \frac{1}{ne^{n^2+1}}
\]
This mathematical expression represents an infinite series where the general term is given by \(\frac{1}{n e^{n^2 + 1}}\). The summation starts from \(n=1\) and extends to infinity.
#### 13e)
\[
\sum_{n=1}^{\infty} \frac{(-1)^n 3^{n+2}}{(n+1)!}
\]
This infinite series involves a term that includes an alternating sign denoted by \((-1)^n\), which causes the terms to alternate between positive and negative. The general term is \(\frac{(-1)^n 3^{n+2}}{(n+1)!}\), and the series summation begins from \(n=1\) and continues to infinity.
#### 13f)
\[
\sum_{n=1}^{\infty} \left( \frac{3n^2 + n + 4}{2n^2 + 1} \right)^n
\]
The third expression represents an infinite series with the general term \(\left( \frac{3n^2 + n + 4}{2n^2 + 1} \right)^n\). This series also starts from \(n=1\) and extends indefinitely.
### Detailed Explanation
- **Infinite Series**: An infinite series is the sum of the terms of an infinite sequence. The expressions shown above involve different types of infinite series, featuring variable components in the terms.
- **Alternating Series**: Some series alternate in sign (e.g., 13e), indicated by the \((-1)^n\) factor, which means the series adds and subtracts terms successively.
These series are often studied in mathematical analysis and have applications in various fields such as physics, engineering, and finance. Analyzing the convergence or divergence of these series is a fundamental aspect of understanding their behavior and application.
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