10.4 Integral Test for Convergence

The integral test applies to a series of the form ${\sum_{n=1}^{\infty} a_n}$ when a_n=f(n) for a function f(x) that is positive, continuous, and decreasing for ${x\ge1}$. Under those conditions, the series and the improper integral ${\int_1^{\infty} f(x)\,dx}$ either both converge or both diverge. This test works because the integral and the series can be compared using areas of rectangles under a decreasing curve.

Before using the integral test, you should always check the hypotheses: positivity, continuity, and decreasing behavior. If those are satisfied, then you evaluate the improper integral instead of the series itself. If the integral converges, the series converges. If the integral diverges, the series diverges. This is especially useful for series whose terms come from algebraic or logarithmic expressions that are easier to study as functions.