Definition. Let f be a function which is continuous on the closed interval [a, ¥). We define

If this limit exists and is finite then we say that the integral is convergent; otherwise, we say that the integral is divergent.

Definition. Let f be a function which is continuous on the closed interval (¥, b]. We define

If this limit exists and is finite then we say that the integral is convergent; otherwise, we say that the integral is divergent.

Definition. Let f be a function which is continuous for all real numbers. If, for some real number c, both of

and

are convergent, then we define

and we say that the integral is convergent; otherwise, we say that the integral is divergent.

Comparsion Theorem. Let f and g be functions which are continuous on the closed interval [a, ¥). Suppose that f(x) > g(x) > 0 for all x in [a, ¥).
  1. If is convergent then is also convergent.

  2. If is divergent then is also divergent.