Very Quick Question: Which Convergence Test to Use?

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Homework Help Overview

The discussion revolves around determining the appropriate convergence test for the series sum from n=2 to infinity of ln(n)/n^2. Participants explore various convergence tests, particularly the comparison test and limit comparison test, while expressing uncertainty about the best approach given the presence of the natural logarithm.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to identify the best convergence test and questions how to select the right comparison for the tests. Some participants discuss the implications of their reasoning and the limits they have calculated, while others suggest applying the integral test and combining known facts about convergence.

Discussion Status

The discussion is active, with participants sharing their thoughts on different convergence tests and questioning their reasoning. Some guidance has been offered regarding the integral test and the comparison test, but there is no explicit consensus on the best approach yet.

Contextual Notes

Participants express difficulty in deciding which convergence tests to apply, particularly when dealing with logarithmic functions. There is a mention of potential mistakes in reasoning and a general uncertainty about the implications of their calculations.

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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Thanks a lot! I made a mistake in reasoning. I appreciate it.
 
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?
 
Really the best way to do this is to apply the integral test.
 
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?
 

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