Very Quick Question: Which Convergence Test to Use?

[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

 

[student45]

Thanks a lot! I made a mistake in reasoning. I appreciate it.

 

[student45]

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?

 

[courtrigrad]

Really the best way to do this is to apply the integral test.

 

[StatusX]

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?