Integral test

Statement

The integral test is a test that allows us to relate the convergence of a series to that of an improper integral. Explicitly, it says that for certain kinds of series, whether or not they converge can be determined by figuring out whether or not an improper integral converges.

Explicitly, suppose $f$ is a function whose domain of definition contains $[x_{0},\infty )$ such that $f$ is continuous, non-negative, and non-increasing (i.e., monotonically, but not necessarily strictly, decreasing) on $[x_{0},\infty )$. Suppose also that $f$ is defined for all nonnegative integers greater than or equal to some integer $k_{0}$. Then:

$\sum _{k=k_{0}}^{\infty }f(k)$ converges $\iff$ $\int _{x_{0}}^{\infty }f(x)\,dx$ converges

Related facts

• Degree difference test combines the integral test with the computation of integrals for power functions.
• Rules for determining interval of convergence sometimes uses the integral test to determine convergence at the endpoints of the interval of convergence.