I have been trying to figure out what the general conditions for absolute convergence of a series are. I was trying to figure this out either working backwards from the ratio test or by considering integrals. Altough I do not know much about what constitutes a 'proof', I believe that using the integral idea the validity of any conditions for absolute convergence I would find would be immediately apparent and the proof itself would be from integral consideration. On the other hand working back from the ratio test would give me some conditions that the ratio test effectively tests for, but it would not give any sort of a proof.
Anyhow, with regards to the ratio test, the only thing I can see that it suggests is that if, after a finite number of terms, the ratio of each successive term is less that one (i.e. series is monotonic decreasing), then we have absolute convergence. This doesn't seem right to me, because the condition for convergence of an alternating series is not only that the terms are monotonic decreasing but also that they tend to 0. So the alternating series test has stricter rules for convergence than what I deduced from the ration test, which makes no sense to me as any alternating series will converge faster than te series formed from absolute terms (if the series of absolute terms converges at all, that is).
So then I tried to consider integrals. If I have a series of a function, then if the integral of the function from some finite x to infinity is finite, and the series 'lies below' the function when drawn out, then the series will converge. So then I was trying to figure out how I would know that the limit to infinity of the integral of some function is fnite. I was considering that first it must be a decreasing function (hence giving me again the condition the ratio of successive terms being less that 1 in the limit as n goes to infinity, as by the ratio test). But there seems to be something else going on here too: it seems that the gradient (which is negative), must also be increasing to 0 fast enough, which hints to me some constarint on the value of the second derivative. I'm not quite sure though...
Any help would be much appreciated!