The integral test applied to the harmonic series. Since the area under the curve y = 1/x for x ∈ [1, ∞) is infinite, the total area of the rectangles must be infinite as well.
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
(1)
for the infinite series.
Note that if the function is increasing, then the function is decreasing and the above theorem applies.
Proof[]
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.
The monotonous function is continuousalmost everywhere. To show this, let . For every , exists by the density of a so that . Note that this set contains an opennon-empty interval precisely if is discontinuous at . We can uniquely identify as the rational number that has the least index in an enumeration and satisfies the above property. Since is monotone, this defines an injectivemapping and thus is countable. It follows that is continuousalmost everywhere. This is sufficient for Riemann integrability.[1]
Since f is a monotone decreasing function, we know that
and
Hence, for every integer n ≥ N,
(2)
and, for every integer n ≥ N + 1,
(3)
By summation over all n from N to some larger integer M, we get from (2)
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 numberk, the series
(4)
still diverges (cf. proof that the sum of the reciprocals of the primes diverges for k = 1) but
(5)
converges for every ε > 0. Here lnk denotes the k-fold composition of the natural logarithm defined recursively by
Furthermore, Nk denotes the smallest natural number such that the k-fold composition is well-defined and lnk(Nk) ≥ 1, i.e.
Whittaker, E. T., and Watson, G. N., A Course in Modern Analysis, fourth edition, Cambridge University Press, 1963. (§ 4.43) ISBN0-521-58807-3
Ferreira, Jaime Campos, Ed Calouste Gulbenkian, 1987, ISBN972-31-0179-3
^Brown, A. B. (September 1936). "A Proof of the Lebesgue Condition for Riemann Integrability". The American Mathematical Monthly. 43 (7): 396–398. doi:10.2307/2301737. ISSN0002-9890. JSTOR2301737.