# Thread: How do I show that this sum converges?

1. ## How do I show that this sum converges?

I'm trying to use the integral test but I have a little problem. I don't know if I am allowed to use l'hopital's rule in integrals because having ln(n)/n^3 would give me 1/3n^3 after applying l'hopital's rule and I can show that 1/n^3 is convergent using the integral test and the comparison test as well however I have a feeling that I am not allowed to use l'hopital's rule in this case, so...what do I do?

Any input would be greatly appreciated!

2. Originally Posted by s3a
I'm trying to use the integral test but I have a little problem. I don't know if I am allowed to use l'hopital's rule in integrals because having ln(n)/n^3 would give me 1/3n^3 after applying l'hopital's rule and I can show that 1/n^3 is convergent using the integral test and the comparison test as well however I have a feeling that I am not allowed to use l'hopital's rule in this case, so...what do I do?

Any input would be greatly appreciated!
That series is a p-series. You should what you know about p series to solve this series.

3. If by that you mean knowing that the sum from 1 to inf of 1/n^3 is convergent then I know that but how does that help me given that 1/n^3 < ln(n)/n^3 ?

4. What is the original problem conditions?

$\frac{1}{n^3}$

This?

5. you should know that $n>ln(n)$

so

$\frac{ln(n)}{n^3} \leq \frac{n}{n^3} = \frac{1}{n^2}$

so ...?