Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![1. Given the series
\[
\sum_{n=1}^{\infty} \frac{3n^2 - 2}{\sqrt{n^8 + n^3 + n + 4}}:
\]
(a) Let \( a_n = \frac{3n^2 - 2}{\sqrt{n^8 + n^3 + n + 4}} \). Define a series \( b_n \) with which to compare it.
(b) Plot the first 50 terms of \( a_n \) and \( b_n \) on the same graph to determine which is larger. If the graph is not clear, use the logical test \( a_n < b_n \) to test the logical value comparing each term.
(c) State whether \( \sum_{n=1}^{\infty} b_n \) converges or not (you should be able to tell by looking), and state whether any conclusion can be made about the convergence of \( \sum_{n=1}^{\infty} a_n \) as a result.
(d) If (c) is inconclusive, determine whether \( \frac{a_n}{b_n} \) converges or not, and state your conclusion about the convergence of \( \sum_{n=3}^{\infty} a_n \) (NOTE: If you still cannot conclude anything, start over with a different \( b_n! \)).
### Explanation of the Task
- **Part (a):** You need to define a comparison series \( b_n \) for the given series \( a_n \).
- **Part (b):** Create a plot with the first 50 terms of \( a_n \) and \( b_n \) to visually compare their sizes.
- **Part (c):** Determine if the series \( \sum b_n \) converges, and then decide if this helps in concluding the behavior of the series \( \sum a_n \).
- **Part (d):** Use the ratio \( \frac{a_n}{b_n} \) if part (c) is not conclusive, to further explore convergence. If this step also fails, begin with a new series \( b_n \).](https://content.bartleby.com/qna-images/question/b529e233-194e-4115-ab2f-ea8d2edde02a/c6d79c11-9212-4624-b4ff-e9b260870e84/gvm8iff_processed.png)
Transcribed Image Text:1. Given the series
\[
\sum_{n=1}^{\infty} \frac{3n^2 - 2}{\sqrt{n^8 + n^3 + n + 4}}:
\]
(a) Let \( a_n = \frac{3n^2 - 2}{\sqrt{n^8 + n^3 + n + 4}} \). Define a series \( b_n \) with which to compare it.
(b) Plot the first 50 terms of \( a_n \) and \( b_n \) on the same graph to determine which is larger. If the graph is not clear, use the logical test \( a_n < b_n \) to test the logical value comparing each term.
(c) State whether \( \sum_{n=1}^{\infty} b_n \) converges or not (you should be able to tell by looking), and state whether any conclusion can be made about the convergence of \( \sum_{n=1}^{\infty} a_n \) as a result.
(d) If (c) is inconclusive, determine whether \( \frac{a_n}{b_n} \) converges or not, and state your conclusion about the convergence of \( \sum_{n=3}^{\infty} a_n \) (NOTE: If you still cannot conclude anything, start over with a different \( b_n! \)).
### Explanation of the Task
- **Part (a):** You need to define a comparison series \( b_n \) for the given series \( a_n \).
- **Part (b):** Create a plot with the first 50 terms of \( a_n \) and \( b_n \) to visually compare their sizes.
- **Part (c):** Determine if the series \( \sum b_n \) converges, and then decide if this helps in concluding the behavior of the series \( \sum a_n \).
- **Part (d):** Use the ratio \( \frac{a_n}{b_n} \) if part (c) is not conclusive, to further explore convergence. If this step also fails, begin with a new series \( b_n \).
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