Question

How to determine if a function is convergent or divergent?

Answer 1

Consider a function f(x) which exhibits a Type I or Type II behavior on the interval [a,b] (in other words, the integral  is improper). We saw before that the this integral is defined as a limit. Therefore we have two cases:           

          1the limit exists (and is a number), in this case we say that the improper integral is convergent;                                                                    2the limit does not exist or it is infinite, then we say that the improper integral is divergent.                                                                                                                                

If the improper integral   is split into a sum of improper integrals (because f(x) presents more than one improper behavior on [a,b]), then the integral converges if and only if any single improper integral is convergent.