Please HELP. Test tomorrow!
Convergence of Series
I have
sum(2 to inf) of 1/(ln(n))^2

I'm fairly certain it diverges. I'm not sure what test to use to show that it diverges though. Any thoughts?

Question

Please HELP. Test tomorrow!
Convergence of Series
I have
sum(2 to inf) of 1/(ln(n))^2

I'm fairly certain it diverges. I'm not sure what test to use to show that it diverges though. Any thoughts?

tkhunny · Answer

Ponder the relationship between $$f(x) = ln(x)$$ and $$g(x) = \sqrt{x}$$

anonymous · Answer

I think just compare it to 1/n....

anonymous · Answer

what were you saying @tkhunny  ?

anonymous · Answer

I don't get it... sqrt(x)?

tkhunny · Answer

$$ln(x) < \sqrt{x}$$ For x > 1, they are both positive, so... $$\left(ln(x)\right)^{2} < x$$.  Are we getting anywhere?

tkhunny · Answer

Whoops, for x > 2 and x an integer, they are both greater than 1.  That's better.

anonymous · Answer

guess we're saying the same thing... compare 1/(lnx)^2 to 1/x...

tkhunny · Answer

Well, we may both be saying it, but neither of us has has quite proved it.  What say you, cosmonautics, is the inuendo good enough or do we need a better demonstration?

anonymous · Answer

well, you showed it...  if (lnx)^2 is less than x  then  1/(lnx)^2 is greater than 1/x

1/x diverges...