Determine if the following series is convergent or divergent. Show all work, give thorough explanations, and state what series convergence test you are using to arrive at your conclusion. (Hint: Think about the largest value that sin(n) can be.) 2 + 3 sin(n) 3n+4 Σ n=1
Determine if the following series is convergent or divergent. Show all work, give thorough explanations, and state what series convergence test you are using to arrive at your conclusion. (Hint: Think about the largest value that sin(n) can be.) 2 + 3 sin(n) 3n+4 Σ n=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
![**Determine if the following series is convergent or divergent.**
Show all work, give thorough explanations, and state what series convergence test you are using to arrive at your conclusion.
*(Hint: Think about the largest value that sin(n) can be.)*
\[
\sum_{n=1}^{\infty} \frac{2 + 3 \sin(n)}{3^n + 4}
\]
**Explanation and Analysis:**
To determine whether the series converges or diverges, consider the largest value that \(\sin(n)\) can take, which is 1. Thus, the expression for the terms of the series becomes:
\[
\frac{2 + 3 \cdot 1}{3^n + 4} = \frac{5}{3^n + 4}
\]
We should test the convergence of this series using an appropriate convergence test, such as the Comparison Test, since the terms resemble a geometric series. We compare it to the simpler series:
\[
\frac{5}{3^n}
\]
The series \(\sum_{n=1}^{\infty} \frac{5}{3^n}\) is a geometric series with a common ratio of \(\frac{1}{3}\), which is less than 1, indicating that it converges.
Using the Comparison Test, since \(\frac{5}{3^n + 4} < \frac{5}{3^n}\) for all \(n \geq 1\), the original series \(\sum_{n=1}^{\infty} \frac{2 + 3 \sin(n)}{3^n + 4}\) also converges.
Conclusion: The given series is convergent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf8dd5bb-1368-4a10-9929-9294802d0b74%2F7dea2805-a970-4c31-89b5-44fdd0cfb2ee%2Fa6c169r_processed.png&w=3840&q=75)
Transcribed Image Text:**Determine if the following series is convergent or divergent.**
Show all work, give thorough explanations, and state what series convergence test you are using to arrive at your conclusion.
*(Hint: Think about the largest value that sin(n) can be.)*
\[
\sum_{n=1}^{\infty} \frac{2 + 3 \sin(n)}{3^n + 4}
\]
**Explanation and Analysis:**
To determine whether the series converges or diverges, consider the largest value that \(\sin(n)\) can take, which is 1. Thus, the expression for the terms of the series becomes:
\[
\frac{2 + 3 \cdot 1}{3^n + 4} = \frac{5}{3^n + 4}
\]
We should test the convergence of this series using an appropriate convergence test, such as the Comparison Test, since the terms resemble a geometric series. We compare it to the simpler series:
\[
\frac{5}{3^n}
\]
The series \(\sum_{n=1}^{\infty} \frac{5}{3^n}\) is a geometric series with a common ratio of \(\frac{1}{3}\), which is less than 1, indicating that it converges.
Using the Comparison Test, since \(\frac{5}{3^n + 4} < \frac{5}{3^n}\) for all \(n \geq 1\), the original series \(\sum_{n=1}^{\infty} \frac{2 + 3 \sin(n)}{3^n + 4}\) also converges.
Conclusion: The given series is convergent.
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