Integral test for convergence
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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
is finite. In other words, if the integral diverges, then the series diverges as well.
If the improper integral is finite, then 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
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 (2)
and from (3)
Combining these two estimates yields
Letting M tend to infinity, the bounds in (1) and the result follow.
The harmonic series
Contrary, the series
From (1) 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 f(n) decreases to 0 faster than 1/n but slower than 1/n1+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
Furthermore, Nk denotes the smallest natural number such that the k-fold composition is well-defined and lnk(Nk) ≥ 1, i.e.
- 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
- Ferreira, Jaime Campos, Ed Calouste Gulbenkian, 1987, ISBN 972-31-0179-3