The second one is correct. Now for the first one

if , then it does not converge since the term oscillates more an more wildly.

If , then you can just use the alternating series test. This term goes to 0 as n goes to infinity, so we have that the series converges! The work you have shown is incorrect though, because you cannot factor -1 out of the absolute value. The limit is indeed 1, so it gives no information on convergence.