Series and convergence/divergence

[Niles]

Homework Statement

Determine whether the series converges:

$$\sum\limits_{n = 2}^{\inf } {\frac{{( - 1)^n (n^2 + 1)^{1/2} }}{{n\ln (n)}}}$$

The Attempt at a Solution

Which test must I use? I thought of using the integral test, but it seems a little too hard. Are there other possibilities?

 

[foxjwill]

Try either the ratio or root test. Neither would be very easy to calculate, but...

 

[NateTG]

Well, the $(-1)^n$ should give you an easy test...

BTW: \infty is $\infty$

 

[Niles]

Should I just take:

$$\sum\limits_{n = 2}^{\inf } {\frac{{( - 1)^n (n^2 + 1)^{1/2} }}{{n\ln (n)}}}^{(1/n)}$$ and take the limit of this?

 

[chaoseverlasting]

Its an alternating series, what about the Leibinitz test?

 

[Niles]

I looked at the Leinitz test - on wikipedia they write about the first condition:

"If the sequence a_n converges to 0, and .." - does this mean the limit of a_n for n -> infinity?