Infinite series

[Nakita]

I'm working infinite series problems for a final review and I'm really fuzzy. I have the problem:

Sigma(n=1) 1 / ( 1 + n ( ln n )[sup:28498dsv]2[/sup:28498dsv] )

And am asked to determine whether it converges/diverges. It looks comparable to 1/n , although I'm sure it cannot be as easy as that. Can someone help me, please?

 

[BigGlenntheHeavy]

\(\displaystyle Hint: \ \frac{1}{n} \ > \ \frac{1}{1+n(ln|n|)^{2}} \ and \ since \ \frac{1}{n} \ diverges, \ this \ tells \ us \ nothing.\)

 

[Nakita]

But the denominator will always be bigger than the numerator, similar to 1/n, right? Wouldn't that make it divergent as well?

 

[chrisr]

not necessarily, Nakita.

Have a look at this link
http://plus.maths.org/issue19/features/infseries/

1+1/2+1/4+1/8+.... does converge.
1/n would have to be < your term to be sure your sum diverges.
What about convergence tests?