Thanks for the responses. This is pretty over my head though, so I'm not sure I understand much.

I asked because of a situation in my calculus class. An exam problem was to test for convergence of the sum from n=whatever to infinity of 1/(n(n^(1/2)+10)). I looked at this and said it compares to 1/n^(3/2). Since p>1, it converges. But my professor marked me wrong because it's not technically correct unless I do the limit comparison test. I'm left wondering why I can't just say it converges since doesn't any function have to converge when the fastest growing term is an exponent and its ratio in the denominator compared to numerator is >1?

I mean, even though technically a p-series is the form 1/n^p, I can't find any more complex functions where p is the fastest growing value and is >1 yet the series doesn't converge. This is technically not p-series: "sum of n=1 to infinity of 1/(n^(100/99)-1000000000000000000000000000000000000000000000000n )", but it still converges on p-series logic. Is there some point where I add enough zeroes to the n^1 term that it starts mattering?

I tried asking my professor but he wasn't understanding my question and almost seemed to be getting mad that I didn't understand why I can't just find the simplest way to get the right answer instead of doing what mathematicians do and rigorously prove it for the sake of rigorous proof, or whatever.