Does this sequence diverge or converge? If it converges, find the limit-

I know the harmonic sequence diverges, but I've taken natural logs of both sides and get stuck, at (1/n)(ln 1 - ln n) = ln L where L is the original limit...

Also,

can this be shown to converge to 0 using the squeeze theorem for sequence convergence? (note that you can move csc to the numerator as )

(both sequences are from n=1 to infinity for the natural numbers)