Convergence of a very simple series

In summary, the conversation discusses a series involving the function a_n = (n^2)/(1-n^3) and how to determine if it converges using standard convergence tests. The speaker hypothesizes that the series may diverge due to a_n looking like 1/n, but is unsure of how to verify this. They also mention wanting to avoid using the integral test. The conversation ends with the suggestion to rewrite the series as S = -s and determine if S diverges or converges with s.
  • #1
Benny
584
0
Hi, there's a really simple looking series that I don't know how to deal with.
[tex]
\sum\limits_{n = 2}^\infty {\frac{{n^2 }}{{1 - n^3 }}}
[/tex]
How would I determine whether or not this series converges by using some standard convergence tests? If a_n = (n^2)/(1-n^3) then the numerator is always positive while the denominator is always negative so that a_n is always negative. So I can't think of a way to use the comparison test, limit comparison test etc. Since a_n looks like 1/n I have a feeling that the series diverges. But I can't think of any tests to use to verify whether or not my hypothesis is correct. At first I thought about using the absolute convergence test but if a series is not absolutely convergent, it can still be convergent so that didn't really help.
Can someone help me with this one?

Note: I would like to be able to do this without the integral test if possible.
 
Last edited:
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  • #2
Rewrite your series S as: [itex]S=-s, s=\sum_{n=2}^{\infty}\frac{n^{2}}{n^{3}-1}[/tex]
S diverges or converges with s.
 
  • #3
Oh ok, thanks for the help arildno.
 

FAQ: Convergence of a very simple series

1. What is the definition of convergence of a very simple series?

The convergence of a very simple series refers to the behavior of the terms in a sequence as the number of terms approaches infinity. It is a measure of whether the terms in a series approach a finite sum or diverge to infinity.

2. How can I determine if a very simple series is convergent or divergent?

There are multiple tests that can be used to determine the convergence or divergence of a very simple series, such as the comparison test, ratio test, and root test. These tests involve analyzing the behavior of the terms in the series and comparing them to known convergent or divergent series.

3. Can a very simple series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is convergent, it means that the terms in the series approach a finite sum as the number of terms approaches infinity. If a series is divergent, it means that the terms in the series approach infinity as the number of terms approaches infinity.

4. What is the significance of the convergence of a very simple series?

The convergence of a very simple series is important in mathematics as it allows for the analysis and prediction of the behavior of infinite processes. It also has applications in various fields such as physics, engineering, and economics.

5. Are there any real-world examples of the convergence of a very simple series?

Yes, there are many real-world examples of the convergence of a very simple series. For instance, the calculation of compound interest in financial investments involves the convergence of a very simple series. Additionally, the estimation of pi using the Leibniz formula involves the convergence of a very simple series.

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