Can someone prove this series converges?

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In summary, the conversation discusses different approaches for testing the convergence of a series, specifically the series \sum_{n=3}^\infty \frac{3}{n^2 - 4}. The speakers mention using the integral test and the limit comparison test, and also discuss the fact that the limit comparison test may be easier to use in this case. They also mention using the comparison test and the fact that the series \frac{3}{n^2-4} is of order \frac{1}{n^2}.
  • #1
[tex]
\sum_{n=3}^\infty \frac{3}{n^2 - 4}.
[/tex]

All the standard approaches I've used have failed. I think the integral test will work, but it's a bit messy.
 
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  • #2
Well it's clearly equivalent to testing for convergence of:
[tex]\sum_{n=3}^\infty \frac{4}{n^2-4}[/tex]
so let's do this instead (the constants turn out nicer). We note:
[tex]\frac{4}{n^2-4} = \frac{1}{n-2}-\frac{1}{n+2}[/tex]
You can use this and a standard telescopping argument to show that it converges.
 
  • #3
Not actually doing this, but couldn't you do the limit comparison test with 1/n^2? Maybe it's not as easy as it seems.
 
  • #4
yuechen said:
Not actually doing this, but couldn't you do the limit comparison test with 1/n^2? Maybe it's not as easy as it seems.

Yes this would work just as well. I used my approach mainly because it also gives you the actual limit (whether you care for it or not), and it uses more elementary tools.

I guess the limit comparison test is actually a bit easier to carry out now that you point it out and it may have been the intended solution.
 
  • #5
As already pointed out, you could use the comparison test, and the fact that :


[tex]\frac{3}{n^2-4}={\cal O}\left(\frac{1}{n^2}\right)[/tex],

for n>k (in this case k=4 would work).
 
Last edited:

1. Can someone prove that a series converges using the ratio test?

Yes, the ratio test is one method that can be used to prove the convergence of a series. It involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges.

2. What is the difference between absolute and conditional convergence?

Absolute convergence means that the series converges when the absolute values of the terms are used. Conditional convergence means that the series converges when the terms are used without absolute values, but it may not converge if the absolute values are used.

3. How can I prove that a series converges using the integral test?

The integral test states that if a series is positive, continuous, and decreasing, then it converges if and only if the corresponding improper integral converges. So, to prove convergence using the integral test, you would need to evaluate the corresponding integral.

4. Can a series converge if its terms do not approach zero?

Yes, it is possible for a series to converge even if its terms do not approach zero. An example of this is the alternating harmonic series, which converges to ln(2) even though its terms do not approach zero.

5. What is the difference between convergence and divergence of a series?

A series is said to converge if the sum of its terms approaches a finite number as the number of terms increases. On the other hand, a series is said to diverge if the sum of its terms does not approach a finite number as the number of terms increases.

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