1. The problem statement, all variables and given/known data I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the riemann sums at n = 1 and makes them all circumscribed rectangles - which make sense to me. The other method starts off the riemann sums at n = 0 and makes them all inscribed rectangles. Can anybody explain to me why each one of these work? Is one right and the other one wrong? 2. Relevant equations 3. The attempt at a solution I've reviewed riemann sums in general, but I'm confused as to why two different methods are used in the theory behind the integral test.