 The most powerful of the convergence test that we have is known as the integral test. Suppose I have a series of positive terms where our terms are given by some function where the function is eventually decreasing. So let's focus on the section where the function is eventually decreasing. Take some sufficiently large k. Consider the area under y equals f of x and above the x-axis over the interval from k to infinity. If we take the left rectangles of width 1, we find the area of each rectangle is f of n times 1. Or, since a sub n is f of n, then the area of this rectangle is going to be a sub n. And because our function is decreasing, these left rectangles will form an upper bound for the area. And this means that the integral from k to infinity of f of x dx will be less than or equal to the series sum from k to infinity of an. Now consider the area under y equals f of x and above the x-axis over the interval from k minus 1 to infinity. In other words, we're going to extend that region back a little bit towards the y-axis. Now suppose we take the right rectangles of width 1. The area of each rectangle is f n times 1, or an. And the sum of the right rectangles is going to be a lower bound for the area. And so the sum from n equals k to infinity of our an is less than or equal to the integral from k minus 1 to infinity. And what this means is that the series sum from k to infinity is bounded from below by one improper integral and bounded above by another improper integral. We can use the additivity property of the definite integral to rewrite these bounds. And since the integral from k minus 1 to k of f of x dx must be finite, this suggests the following result. Let our series be an eventually positive series where the terms an are given by f n for some eventually decreasing function f of x. Then our series converges if and only if the improper integral converges. And while strictly speaking this improper integral does need a starting point, as long as our series is eventually decreasing, it doesn't actually matter where we start. As you might expect, there's a catch. The integral test is the most powerful test available. It applies to any eventually decreasing series of positive terms and it always returns a decision on the convergence or divergence of a series. However, there's one important limitation. We have to evaluate an integral and evaluating the improper integral is not always possible or practical. For example, say we want to determine the convergence or divergence of the series 1 over n squared. So first of all, we note that f n equals 1 over n squared is positive and eventually decreasing. So we can apply the integral test. So we'll evaluate our integral from some place to infinity of 1 over x squared dx. Since that's an improper integral, we have to rephrase this as a limit. We'll find the antiderivative. We'll evaluate it. We'll take the limit. And since the integral converges 1 over k is a finite value, then our series also converges.