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
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 that, for every natural number, the series
still diverges 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.