Expert: Jack Cheng - 12/1/2006

Question
I was wondering if you could help me with this problem please.
                               infinity
Determine if (squareroot n/log n)
                               n=2

converges or not. If it does, find the limit.
Thanks,
Kathrine  

Answer
Hi Kathrine,

I am assuming you mean the sum of sqrt(n)/log n from n=2 to infinity, and log is the natural log.

For this to be convergent, lim n->infinity sqrt(n)/log n must be 0 (by the Divergence Test). To find out if it does, let's use L'Hopital's Rule.

[VIEW WITH FIXED-SIZE FONT LIKE COURIER]

sqrt(n)    1/(2sqrt(n))         n
------- =  ------------  = -----------
log n           1/n         2 sqrt(n)

Using L'Hopital's Rule again:

   n            1          
--------- = ---------- = sqrt(n) -> infinity
2 sqrt(n)    1/sqrt(n)

Because it doesn't approach zero as n approaches infinity, the summation is divergent.

I hope this helps,
Jack