Integral test for convergence
In mathematics, 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  and a non-negative function  defined on the unbounded interval , on which it is monotone decreasing. Then the infinite series

converges to a real number if and only if the improper integral

is finite. In other words, if the integral diverges, then the series diverges as well.

Remark
If the improper integral is finite, then the proof also gives the lower and upper bounds

for the infinite series.

Proof
The proof basically uses the comparison test, comparing the term  with the integral of  over the intervals
 and , respectively.

Since  is a monotone decreasing function, we know that

and

Hence, for every integer ,

and, for every integer ,

By summation over all  from  to some larger integer , we get from ()

and from ()

Combining these two estimates yields

Letting  tend to infinity, the bounds in () and the result follow.

Applications
The harmonic series

diverges because, using the natural logarithm, its antiderivative, and the fundamental theorem of calculus, we get

Contrary, the series

(cf. Riemann zeta function)
converges for every , because by the power rule

From () we get the upper estimate

which can be compared with some of the particular values of Riemann zeta function.

Borderline between divergence and convergence
The above examples involving the harmonic series raise the question, whether there are monotone sequences such that  decreases to 0 faster than  but slower than  in the sense that

for every , and whether the corresponding series of the  still diverges. Once such a sequence is found, a similar question can be asked with  taking the role of , and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series.

Using the integral test for convergence, one can show (see below) that, for every natural number , the series

still diverges (cf. proof that the sum of the reciprocals of the primes diverges for ) but

converges for every . Here  denotes the -fold composition of the natural logarithm defined recursively by

Furthermore,  denotes the smallest natural number such that the -fold composition is well-defined and , i.e.

using tetration or Knuth's up-arrow notation.

To see the divergence of the series () using the integral test, note that by repeated application of the chain rule

hence

To see the convergence of the series (), note that by the power rule, the chain rule and the above result

hence

and () gives bounds for the infinite series in ().