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Calculus |
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Specialized calculi |

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.

Consider an integer *N* and a non-negative function *f* defined on the unbounded interval [*N*, ∞), 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.

The proof also gives the lower and upper bounds

for the infinite series.

The proof basically uses the comparison test, comparing the term *f*(*n*) with the integral of *f* over the intervals [*n* − 1, *n*) and [*n*, *n* + 1), respectively.

Since *f* is a monotone decreasing function, we know that

and

Hence, for every integer *n* ≥ *N*,

**(**

and, for every integer *n* ≥ *N* + 1,

**(**

By summation over all *n* from *N* to some larger integer *M*, we get from (**1**)

and from (**2**)

Combining these two estimates yields

Letting *M* tend to infinity, the result follows.

The harmonic series

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

Contrary, the series

(cf. Riemann zeta function) converges for every *ε* > 0, because by the power rule

The above examples involving the harmonic series raise the question, whether there are monotone sequences such that *f*(*n*) decreases to 0 faster than 1/*n* but slower than 1/*n*^{1+ε} in the sense that

for every *ε* > 0, and whether the corresponding series of the *f*(*n*) still diverges. Once such a sequence is found, a similar question can be asked with *f*(*n*) taking the role of 1/*n*, 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 *k*, the series

**(**

still diverges (cf. proof that the sum of the reciprocals of the primes diverges for *k* = 1) but

**(**

converges for every *ε* > 0. Here ln_{k} denotes the *k*-fold composition of the natural logarithm defined recursively by

Furthermore, *N*_{k} denotes the smallest natural number such that the *k*-fold composition is well-defined and ln_{k}(*N*_{k}) ≥ 1, i.e.

using tetration or Knuth's up-arrow notation.

To see the divergence of the series (**3**) using the integral test, note that by repeated application of the chain rule

hence

To see the convergence of the series (**4**), note that by the power rule, the chain rule and the above result

hence

- Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.3) ISBN 0-486-60153-6
- Whittaker, E. T., and Watson, G. N.,
*A Course in Modern Analysis*, fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN 0-521-58807-3