How do you test the alternating series #Sigma (n(-1)^(n+1))/lnn# from n is #[2,oo)# for convergence?
is not convergent.
A necessary condition for any series to converge is that:
which also implies:
In our case:
so the series is not convergent.
We can also see that taking only the terms of even order:
while for terms of odd order:
so the series is irregular.