Does the series Sigma(n=1 to infinity) (-1)^n ln[(3n+1)/(3n-2)] converge?

Answer 1

Yes. Since the sequence goes to zero (lim ln[(3n+1)/(3n-2)]=ln[lim(3n+1)/(3n-2)]=ln[3/3]=ln[1]=0), and the sequence is decreasing on the interval [1,infinty) (take the derivative of f(x)=ln[(3n+1)/(3n-2)] and see it is less than zero for all numbers bigger than 1), by the alternating series test the sum coverages!