I have a big ol' ugly mid-term tomorrow on series convergence. Tonight I've been clunking through all the definitions and endless theorems and have become curious about a couple of things.
The integral test for convergence works in the case of functions that are positive, decreasing, and continuous, right? Well, my question here is why not alternating, or discontinuous? Hey, at least I get the decreasing part...right? Anyway:
1) Take the case of a function that is asymptotic at some number, let it be c. And c is an integer, as they often are. In that case, why could we not evaluate the improper integral from 0 to c, and c to infinity? We do allow an asymptote, and an infinite interval in many functions for which we evaluate integrals, do we not?
2) In the case of an alternating series, I believe that the series sum is the absolute value of sum a of n as n goes to infinity, right? (Please correct me if i am wrong, and forgive my lack of latex chops) Hence, my question is why would that sum not be the abs value of the function's integral?
OK, that's it for now.