How would i test this series for convergence or divergence? and what would be the procedures?

Question

summation n=1-infiniti
{[(-1)^(n-1)][ln(n)]}/n

Answer 1

take first abs val
a_n = ln/n >1/n divergent
1) lim ln/n =0 (L´Hôpital)
2) a_n is monotone decreasing
Take f(x) = lnx/x so f´(x= = 1/x^2*( 1-lnx) which is <0 for x>e
so f(x) is decreasing and so a_n
so(-1)^(n-1)* a_n is convergent

Answer 2

♠ for this sign alternating series we have condition
|a[n+1]| < |a[n]|, while a[n] → 0; good for convergence!
♣ meanwhile ∑ln(n)/n diverges as it is worse than
harmonic series, since ln(n)/n > 1/n;
♠ thus ∑(-1)^(n-1) ln(n) / n converges conditionally!