The integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

## Statement of the test

Consider an integer

Although we won't go into the details, the proof of the test also gives the lower and upper bounds:

for the infinite series.

## Applications

The harmonic series

On the other hand, the series

## Integral Test

The integral test applied to the harmonic series. Since the area under the curve

The above examples involving the harmonic series raise the question of whether there are monotone sequences such that

and:

for every