it seems that you are starting to understand it better.
your first example is correct. if the series that converges is greater than the one in question, it converges.
however, your second example is incorrect. if we compare the series in question (the "1st equation") to a series that converges, but it turns out that the 1st equation is greater than the converging one, it doesnt necessarily mean that it converges OR diverges.
think about both alternatives. you could have the "1st equation" still converge AND be greater than another converging series. or on the other hand, you could have the "1st equation" be a DIVERGENT series and STILL be greater than the convergent series. The problem is that we cannot determine what its behavior is with this certain series. another one must be found that can show it converges (i.e. one that converges itself.)
so no, if a series is greater than another convergent series, it does not necessarily mean it diverges.
however, if you showed that a series (lets stick with "1st equation") is greater than a DIVERGENT series, then you know for a fact that it diverges. for example, if we say that series A is divergent, then any series BIGGER than a divergent series could not ever converge.