# Integral Test: What should I compare this series with to prove it's convergence?

• theBEAST
In summary, the conversation discusses using the Integral Test to prove the convergence of a given problem. The solution involves comparing the given problem with a similar one, \frac{\sqrt{2n+2n}}{n^2}, to further support its convergence.
theBEAST

## Homework Statement

Here is the problem:
http://dl.dropbox.com/u/64325990/HW%20Pictures/integraltest.PNG

## The Attempt at a Solution

I know it is convergent because it is very similar to 1/n^1.5 which is convergent as well. However what would I compare this with using the Integral Test to prove that it is convergent?

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theBEAST said:

## Homework Statement

Here is the problem:
http://dl.dropbox.com/u/64325990/HW%20Pictures/integraltest.PNG

## The Attempt at a Solution

I know it is convergent because it is very similar to 1/n^1.5 which is convergent as well. However what would I compare this with using the Integral Test to prove that it is convergent?

Compare it with $\frac{\sqrt{2n+2n}}{n^2}$. Then it's even more similar to 1/n^(1.5).

Last edited by a moderator:

## 1. What is the purpose of the Integral Test?

The Integral Test is used to determine the convergence or divergence of an infinite series. It is particularly useful for series that cannot be evaluated using other tests, such as the comparison test or limit comparison test.

## 2. How do I apply the Integral Test to a series?

To apply the Integral Test, you must first determine if the series is positive, decreasing, and continuous. Then, you need to find a function that is greater than or equal to the series and integrate it. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

## 3. Can I use any function to compare with the series?

No, the function used for comparison must be positive, decreasing, and continuous on the interval of the series. It is also helpful to choose a function that is easy to integrate.

## 4. Is the Integral Test always reliable?

The Integral Test is a reliable test for determining convergence or divergence, but it is not foolproof. There are some cases where the integral may converge, but the series may actually diverge. This is why it is important to also consider other tests and approaches when determining the convergence of a series.

## 5. Can the Integral Test be used for alternating series?

No, the Integral Test can only be used for series with positive terms. For alternating series, the Alternating Series Test should be used instead.

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