Very Quick Question: Which Convergence Test to Use?

In summary, the best convergence test to use for the sum from n=2 to infinity of ln(n)/n^2 is the integral test. This is because the series ln(n)/n^e goes to zero as n goes to infinity for any e>0, and the series 1/n^(1+e) converges for any e>0. The comparison test can also be used in this case. However, there is no hard and fast rule for determining which test to use, so it is important to carefully consider the properties of the series and choose the most appropriate test.
  • #1
student45
What is the best convergence test to use for the sum from n=2 to infinity of ln(n)/n^2? The comparison test and limit comparison test both probably work... but what is the right comparison for each of these tests? I have always had a hard time deciding which tests to use, especially when the natural log is thrown in there. Is there any hard and fast rule for determining what to do with this type of problem?

Thanks a lot.

-student45
 
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  • #2
Thanks a lot! I made a mistake in reasoning. I appreciate it.
 
  • #3
What are the steps for that?

I end up with lim[n->inf.] (1/n) --> 0. But since 1/n diverges generally, does this even tell me anything?
 
  • #4
Really the best way to do this is to apply the integral test.
 
  • #5
You have that ln(n)/n^e goes to zero as n goes to infinity for any e>0. You also have that the series 1/n^(1+e) converges for any e>0. Can you combine these facts and use the comparison test?
 

1. What is a convergence test?

A convergence test is a method used to determine whether a series converges or diverges. It helps to determine the behavior of a series as the number of terms increases.

2. Why is it important to know which convergence test to use?

Using the appropriate convergence test can help us determine the convergence or divergence of a series more accurately and efficiently. It also helps us understand the behavior of a series and make predictions about its sum.

3. What are some common convergence tests?

Some common convergence tests include the ratio test, the root test, the comparison test, the limit comparison test, the integral test, and the alternating series test.

4. How do I decide which convergence test to use?

There is no one-size-fits-all approach to choosing a convergence test. It often depends on the specific series and its terms. It is important to understand the conditions and limitations of each test and apply the one that is most suitable for the given series.

5. Is there a foolproof method for determining convergence or divergence?

No, there is no foolproof method for determining convergence or divergence of a series. However, using a combination of convergence tests and understanding the behavior of the series can help us make accurate conclusions about its convergence or divergence.

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