Solving the Conditional Convergence of \((-1)^n/ln(n)\) Series

[frasifrasi]

I am trying to practice for an exam but can't do this question:

does the series $$\((-1)^n/ln(n)$$ from n = 2 to infinity converge abs/conditionally/diverge?

I know if a do an alternating series test, the integral will converge because lim goes to 0 and a(n+1)<an.

But how can I prove that it's conditionally convergent? I did the limit test but it says that it is absolutely convergent, which is not the answer(it is supposed to be conditionally).

Thank you...

 

[Gib Z]

Well it can't be absolutely, because 1/( ln n ) goes to 0 much slower than another well known series that diverges doesn't it?

 

[frasifrasi]

ok, 1/n diverges as is a bigger series, so 1/ln(n) must also converge, right?

 

[SiddharthM]

write it with inequalities and using the definition of "diverge" show that 1/n FORCES 1/lnn to blow up.

 

[frasifrasi]

Sorry I meant, since thesmaller series diverges, 1/ln(n) will also diverge.

 

[HallsofIvy]

And since 1/ln n is a decreasing sequence, the "conditional convergence" part is easy!