Integral test for convergence on arvau

Wikipedia / Integral test for convergence

In mathematics, the integral test for convergence is a method used to test infinite series of monotonic 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 function defined on the unbounded interval , on which it is monotonically decreasing. Then the infinite series

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

is finite. In particular, 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.

is decreasing and the above theorem applies.

to be positive, diverge.

Proof

The proof uses the comparison test respectively.

is continuous almost everywhere. To show this, let

, there exists by the density.

Note that this set contains an open non-empty is discontinuous as the rational number that has the least index in an enumeration is monotone, this defines an injective mapping is countable is continuous almost everywhere. This is sufficient for Riemann integrability.

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

On the other hand, 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 of 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 either (if diverges) or (if converges), 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, let denote 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,