Hey MathIsOhSoHard.
Can you outline the example you are talking about?
If an infinite series is not monotonically decreasing and if it does not approach 0 but instead approaches infinity, would this be sufficient information to conclude that the series is divergent since it doesn't meet the alternating test's requirements?
The nth term test should show that this diverges as long as n > 4 so you can break up the series with n = 1 to 4 and then another series with n = 4 onwards and show that absolute value is greater than 1 for all terms which means it diverges.