Use the integral test to determine if this series converges or diverges

In summary, the conversation discusses the use of the integral test to determine if a given series converges or diverges. The individual seeking help has attempted to find the integral using an online calculator, but has not been successful. Another person suggests using u-substitution and the original individual confirms that they were able to find the integral using this method.
  • #1
Randall
22
0

Homework Statement


Use the integral test to determine if this series converges or diverges: sum from n=1 to infinity of n/(1+(n^2))

Homework Equations


Integral test: a series and it's improper integral both either converge or both diverge

The Attempt at a Solution


see attached - I need help finding the integral. I tried using an online integral calculator, symbolab but it says there is no integral. I'm guessing there is some way to split this up into pieces that I'm not seeing. Please help.
 

Attachments

  • integral_test.jpg
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  • #2
Your proof that it is positive and decreasing isn't sufficient, but the integral is fairly trivial. U-substitution comes to mind.
 
  • #3
Thank you! Yes I used substitution and got it to work :)
 

1. What is the integral test?

The integral test is a method used to determine the convergence or divergence of a series by comparing it to the convergence or divergence of an improper integral. It is based on the idea that if the corresponding integral converges, then the series also converges, and vice versa.

2. When should the integral test be used?

The integral test is typically used for series where the terms involve a rational function or a power of n. It is also useful for series that cannot be easily evaluated using other convergence tests such as the comparison or ratio test.

3. How is the integral test used?

To use the integral test, the first step is to express the series as a function of n. Then, find the corresponding improper integral. If the integral is convergent, then the series is also convergent. If the integral is divergent, then the series is also divergent. If the integral is inconclusive, then the test cannot be used to determine the convergence or divergence of the series.

4. Are there any limitations to the integral test?

Yes, there are some limitations to the integral test. It can only be used for series with positive terms, and the terms must be continuous, decreasing, and positive for all n. Additionally, the test may not work for series with alternating signs or oscillating terms.

5. What is the advantage of using the integral test?

The advantage of using the integral test is that it can provide a conclusive answer for the convergence or divergence of a series, whereas other tests may only give inconclusive results. It also allows for the comparison of more complex series by reducing them to a simple integral.

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