Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Series**
\[
\sum_{n=1}^{\infty} \frac{n}{n^2+1}
\]
is a divergent series. Which of the following test(s) can be used to show its divergence?
(A). The Divergence Test
(B). The Integral Test
(C). The Limit Comparison Test
(D). The Ratio Test
**Explanation:**
In this problem, we are tasked with determining which test(s) can be used to show the divergence of the series \(\sum_{n=1}^{\infty} \frac{n}{n^2+1}\). The available tests are:
- **The Divergence Test**: This test checks if the sequence of terms does not converge to zero, which implies divergence of the series.
- **The Integral Test**: This is useful when you can integrate the function that describes your sequence to determine convergence or divergence.
- **The Limit Comparison Test**: This involves comparing the given series with another series whose convergence is known.
- **The Ratio Test**: Typically used for series whose terms involve factorials, exponentials or powers, checking the growth of terms.
Through these methods, one can evaluate how to demonstrate the divergence effectively.](https://content.bartleby.com/qna-images/question/94d2dcc2-a5ed-45f9-a850-7aa9d120f65f/ff3fc747-22dd-433c-8cdd-f441732faf69/x6eizzl_processed.jpeg)
Transcribed Image Text:**Series**
\[
\sum_{n=1}^{\infty} \frac{n}{n^2+1}
\]
is a divergent series. Which of the following test(s) can be used to show its divergence?
(A). The Divergence Test
(B). The Integral Test
(C). The Limit Comparison Test
(D). The Ratio Test
**Explanation:**
In this problem, we are tasked with determining which test(s) can be used to show the divergence of the series \(\sum_{n=1}^{\infty} \frac{n}{n^2+1}\). The available tests are:
- **The Divergence Test**: This test checks if the sequence of terms does not converge to zero, which implies divergence of the series.
- **The Integral Test**: This is useful when you can integrate the function that describes your sequence to determine convergence or divergence.
- **The Limit Comparison Test**: This involves comparing the given series with another series whose convergence is known.
- **The Ratio Test**: Typically used for series whose terms involve factorials, exponentials or powers, checking the growth of terms.
Through these methods, one can evaluate how to demonstrate the divergence effectively.
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