For the second one, if n is an integer then (-2)^n = 2^n * (-1)^n. The first one is different though and has no easy factorization with (-1)^n (but you can look at complex numbers).
Are you sure the first one can't be rewritten?
I need to show that is either convergent or divergent and since the expression is an alternating series, I should be able to use Leibniz criterion however I need to rewrite it with as a factor.
Are you sure this can't be done?
I've always called it the alternating series test instead of Leibniz criterion - just in case you run across that terminology.
For absolute convergence, you can just say that:
It's true that skeeter's bound is tighter, but you don't need it to prove convergence.