Testing a series for convergence

• a.s
In summary, the conversation is about a student struggling with a math problem involving a series and various series tests. They are having trouble finding a larger series for comparison and have attempted the ratio test with difficult results. The conversation ends with a request for help.
a.s
EDIT: The latex doesn't seem to be working at all... not exactly sure why this is. I can't delete the post, so, uh... never mind, I guess?
EDIT v. 2.0: Yeah, so copying and pasting images from LatexIt is apparently beyond me... Thanks for the help, razored! Right, so, any help anyone can give will be well appreciated!

Homework Statement

Determine whether the following series converges or diverges. If possible, determine the sum of the series exactly. Justify your answer with the proper series test.
$$\sum_{n=2}^\infty{\sqrt{n^3+3}-\sqrt{n^3-3}}$$

Homework Equations

Comparison test, integral test, root test, ratio test, etc.

The Attempt at a Solution

Multiplying by the conjugate
$$\sqrt{n^3+3}+\sqrt{n^3-3}$$
produces the series
$$\sum_{n=2}^\infty{\frac{6}{\sqrt{n^3+3}+\sqrt{n^3-3}}}$$.

My first inclination was to try and find a series which is clearly larger, but I'm having trouble doing that. In previous problems like this that I've seen, there's been an n in the denominator, allowing me to eliminate the terms with square roots and produce a series which is always greater.

In this case, though, I can't, so I gave the ratio test a try, with painful results:
$$\lim_{n\rightarrow\infty}\frac{\sqrt{(n+1)^3+3}+\sqrt{(n+1)^3-3}}{\sqrt{n^3+3}+\sqrt{n^3-3}}$$.
I tried expanding the cubic terms, making it a bit ridiculous.
$$\lim_{n\rightarrow\infty}\frac{\sqrt{n^3+3n^2+3n+4}+\sqrt{n^3+3n^2+3n-2}}{\sqrt{n^3+3}+\sqrt{n^3-3}}$$

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1. What does it mean for a series to converge?

Convergence in a series means that the terms of the series become smaller and smaller as the series progresses, eventually approaching a specific value. This value is known as the limit of the series.

2. How do you test for convergence in a series?

There are several methods for testing convergence in a series, including the ratio test, the root test, and the comparison test. These methods involve comparing the given series to a known convergent or divergent series and using algebraic manipulation to determine the behavior of the given series.

3. What is the ratio test for convergence?

The ratio test is a method for determining the convergence of a series by taking the limit of the absolute value of the ratio between consecutive terms in the series. If this limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive and another method must be used.

4. How does the comparison test work?

The comparison test is a method for determining the convergence of a series by comparing it to a known convergent or divergent series. If the given series is smaller than the known convergent series, it will also converge. If the given series is larger than the known divergent series, it will also diverge.

5. Is it possible for a series to be neither convergent nor divergent?

Yes, it is possible for a series to be neither convergent nor divergent. This can occur when the terms of the series do not approach a specific value or diverge to infinity, but instead oscillate or behave unpredictably.

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