We will use the alternating series test, which says that for some series sum(-1)^na_n, the series converges if a_n is decreasing and lim_(nrarroo)a_n=0.
We have the series sum_(n=1)^oo((-1)^(n+1)ln(n+1))/(n+1). We see that (-1)^(n+1) is the alternating portion so our sequence in question is a_n=ln(n+1)/(n+1).
In order to determine if the series is convergent, we need to determine if the criteria are true.
We can see that ln(n+1)/(n+1) is decreasing by noting that n+1 will increase faster than ln(n+1), so the fraction will get smaller.
You could also graph this, or show that a_(n-1)/a_n<1, which means that the current term is greater than the previous term. If we want, we can show that (ln(n)/(n))/(ln(n+1)/(n+1))<1.
The second criterion is that lim_(nrarroo)ln(n+1)/(n+1)=0. We can do this by again recognizing than n grows far faster than ln(n).
We can also use L'Hôpital's rule to find the limit: lim_(nrarroo)ln(n+1)/(n+1)=lim_(nrarroo)(1/(n+1))/1=0.
Thus, sum_(n=1)^oo((-1)^(n+1)ln(n+1))/(n+1) is a convergent series.
