Determine convergence for a series

[MozAngeles]

Homework Statement

Sum ln n/(ln(ln n)) n=3..infinity?
Im pretty sure it diverges and I am pretty sure to use the limit test but i just don't know what to compare this sum to. Would 1/n be ok. Do i have to justify why they are similar?

ANy help would be nice thanks.

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

Sum ln n/(ln(ln n)) n=3..infinity?
Im pretty sure it diverges and I am pretty sure to use the limit test but i just don't know what to compare this sum to. Would 1/n be ok. Do i have to justify why they are similar?

ANy help would be nice thanks.

Sure, you can use 1/n in the limit comparison test.

If this limit is positive, then your series diverges.
$$\lim_{n \to \infty}\frac{\frac{ln(n)}{ln(ln(n))}}{1/n}$$

 

[Char. Limit]

You could also just see that log(n) is larger than log(log(n)) for basically every n you're using, and so you're summing a sequence of numbers that all are greater than 1. So realistically, you could use a limit comparison with 1.

 

[wisvuze]

even worse, the tail of this series doesn't seem to go to 0, log(x) will grow faster than log(log(x))

 

[Mark44]

You could also just see that log(n) is larger than log(log(n)) for basically every n you're using, and so you're summing a sequence of numbers that all are greater than 1. So realistically, you could use a limit comparison with 1.

even worse, the tail of this series doesn't seem to go to 0, log(x) will grow faster than log(log(x))

These observations suggest that the nth term test for divergence might be appropriate here.

 

[MozAngeles]

thanks i figured it out like two seconds after i posted. haha