The name of the harmonic series derives from the concept of
overtones or harmonics
in music: the
wavelengths of the overtones of a vibrating string are ,,, etc., of the string's
fundamental wavelength. Every term of the harmonic series after the first is the
harmonic mean of the neighboring terms, so the terms form a
harmonic progression; the phrases harmonic mean and harmonic progression likewise derive from music.
Beyond music, harmonic sequences have also had a certain popularity with architects. This was so particularly in the
Baroque period, when architects used them to establish the
floor plans, of
elevations, and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.
in which the terms are all of the positive
unit fractions. It is a
divergent series: as more terms of the series are included in
partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit. Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps.
Two of the best-known are listed below.
One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest
power of two:
Grouping equal terms shows that the second series diverges (because every grouping of convergent series is only convergent):
Because each term of the harmonic series is greater than or equal to the corresponding term of the second series (and the terms are all positive), it follows (by the
comparison test) that the harmonic series diverges as well. The same argument proves more strongly that, for every
Rectangles with area given by the harmonic series, and the hyperbola through the upper left corners of these rectangles
It is possible to prove that the harmonic series diverges by comparing its sum with an
improper integral. Specifically, consider the arrangement of rectangles shown in the figure to the right. Each rectangle is 1 unit wide and units high, so if the harmonic series converged then the total area of the rectangles would be the sum of the harmonic series. The curve stays entirely below the upper boundary of the rectangles, so the area under the curve (in the range of from one to infinity that is covered by rectangles) would be less than the area of the union of the rectangles. However, the area under the curve is given by a divergent
Because this integral does not converge, the sum cannot converge either.
Replacing each rectangle by the next one in the sequence would produce a sequence of rectangles whose boundary lies below the curve rather than above it.
This shows that the partial sums of the harmonic series differ from the integral by an amount that is bounded above and below by the unit area of the first rectangle:
Generalizing this argument, any infinite sum of values of a monotone decreasing positive function of (like the harmonic series) has partial sums that are within a bounded distance of the values of the corresponding integrals. Therefore, the sum converges if and only if the integral over the same range of the same function converges. When this equivalence is used to check the convergence of a sum by replacing it with an easier integral, it is known as the
integral test for convergence.
No harmonic numbers are integers, except for . One way to prove that is not an integer is to consider the highest
power of two in the range from 1 to . If is the
least common multiple of the numbers from 1 to , then
can be rewritten as a sum of fractions with equal denominators
in which only one of the numerators, , is odd and the rest are even, and (when ) is itself even. Therefore, the result is a fraction with an odd numerator and an even denominator, which cannot be an integer. More strongly, any sequence of consecutive integers has a unique member divisible by a greater power of two than all the other sequence members, from which it follows by the same argument that no two harmonic numbers differ by an integer.
Another proof that the harmonic numbers are not integers observes that the denominator of must be divisible by
prime numbers greater than , and uses
Bertrand's postulate to prove that this set of primes is non-empty. The same argument implies more strongly that, except for , , and , no harmonic number can have a
terminating decimal representation. It has been conjectured that every prime number divides the numerators of only a finite subset of the harmonic numbers, but this remains unproven.
Just as the gamma function provides a continuous
interpolation of the
factorials, the digamma function provides a continuous interpolation of the harmonic numbers, in the sense that .
This equation can be used to extend the definition to harmonic numbers with rational indices.
Many well-known mathematical problems have solutions involving the harmonic series and its partial sums.
Solution to the jeep problem for , showing the amount of fuel in each depot and in the jeep at each step
jeep problem or desert-crossing problem is included in a 9th-century problem collection by
Alcuin, Propositiones ad Acuendos Juvenes (formulated in terms of camels rather than jeeps), but with an incorrect solution. The problem asks how far into the desert a jeep can travel and return, starting from a base with loads of fuel, by carrying some of the fuel into the desert and leaving it in depots. The optimal solution involves placing depots spaced at distances from the starting point and each other, where is the range of distance that the jeep can travel with a single load of fuel. On each trip out and back from the base, the jeep places one more depot, refueling at the other depots along the way, and placing as much fuel as it can in the newly placed depot while still leaving enough for itself to return to the previous depots and the base. Therefore, the total distance reached on the th trip is
where is the th harmonic number. The divergence of the harmonic series implies that crossings of any length are possible with enough fuel.
For instance, for Alcuin's version of the problem, : a camel can carry 30 measures of grain and can travel one leuca while eating a single measure, where a leuca is a unit of distance roughly equal to 2.3 kilometres (1.4 mi). The problem has : there are 90 measures of grain, enough to supply three trips. For the standard formulation of the desert-crossing problem, it would be possible for the camel to travel leucas and return, by placing a grain storage depot 5 leucas from the base on the first trip and 12.5 leucas from the base on the second trip. However, Alcuin instead asks a slightly different question, how much grain can be transported a distance of 30 leucas without a final return trip, and either strands some camels in the desert or fails to account for the amount of grain consumed by a camel on its return trips.
block-stacking problem: blocks aligned according to the harmonic series can overhang the edge of a table by the harmonic numbers
block-stacking problem, one must place a pile of identical rectangular blocks, one per layer, so that they hang as far as possible over the edge of a table without falling. The top block can be placed with of its length extending beyond the next lower block. If it is placed in this way, the next block down needs to be placed with at most of its length extending beyond the next lower block, so that the
center of mass of the top two block is supported and they do not topple. The third block needs to be placed with at most of its length extending beyond the next lower block, and so on. In this way, it is possible to place the blocks in such a way that they extend lengths beyond the table, where is the th harmonic number. The divergence of the harmonic series implies that there is no limit on how far beyond the table the block stack can extend. For stacks with one block per layer, no better solution is possible, but significantly more overhang can be achieved using stacks with more than one block per layer.
where denotes the set of prime numbers. The left equality comes from applying the
distributive law to the product and recognizing the resulting terms as the
prime factorizations of the terms in the harmonic series, and the right equality uses the standard formula for a
geometric series. The product is divergent, just like the sum, but if it converged one could take logarithms and obtain
Here, each logarithm is replaced by its
Taylor series, and the constant on the right is the evaluation of the convergent series of terms with exponent greater than one. It follows from these manipulations that the sum of reciprocals of primes, on the right hand of this equality, must diverge, for if it converged these steps could be reversed to show that the harmonic series also converges, which it does not. An immediate corollary is that
there are infinitely many prime numbers, because a finite sum cannot diverge. Although Euler's work is not considered adequately rigorous by the standards of modern mathematics, it can be made rigorous by taking more care with limits and error bounds. Euler's conclusion that the partial sums of reciprocals of primes grow as a
double logarithm of the number of terms has been confirmed by later mathematicians as one of
Mertens' theorems, and can be seen as a precursor to the
prime number theorem.
The operation of rounding each term in the harmonic series to the next smaller integer multiple of causes this average to differ from the harmonic numbers by a small constant, and
Peter Gustav Lejeune Dirichlet showed more precisely that the average number of divisors is (expressed in
big O notation). Bounding the final error term more precisely remains an open problem, known as
Dirichlet's divisor problem.
Graph of number of items versus the expected number of trials needed to collect all items
Several common games or recreations involve repeating a random selection from a set of items until all possible choices have been selected; these include the collection of
trading cards and the completion of
parkrun bingo, in which the goal is to obtain all 60 possible numbers of seconds in the times from a sequence of running events. More serious applications of this problem include sampling all variations of a manufactured product for its
quality control, and the
random graphs. In situations of this form, once there are items remaining to be collected out of a total of equally-likely items, the probability of collecting a new item in a single random choice is and the expected number of random choices needed until a new item is collected is . Summing over all values of from down to 1 shows that the total expected number of random choices needed to collect all items is , where is the th harmonic number.
Animation of the average-case version of quicksort, with recursive subproblems indicated by shaded arrows and with pivots (red items and blue lines) chosen as the last item in each subproblem
quicksort algorithm for sorting a set of items can be analyzed using the harmonic numbers. The algorithm operates by choosing one item as a "pivot", comparing it to all the others, and recursively sorting the two subsets of items whose comparison places them before the pivot and after the pivot. In either its
average-case complexity (with the assumption that all input permutations are equally likely) or in its
expected time analysis of worst-case inputs with a random choice of pivot, all of the items are equally likely to be chosen as the pivot. For such cases, one can compute the probability that two items are ever compared with each other, throughout the recursion, as a function of the number of other items that separate them in the final sorted order. If items and are separated by other items, then the algorithm will make a comparison between and only when, as the recursion progresses, it picks or as a pivot before picking any of the other items between them. Because each of these items is equally likely to be chosen first, this happens with probability . The total expected number of comparisons, which controls the total running time of the algorithm, can then be calculated by summing these probabilities over all pairs, giving
The depleted harmonic series where all of the terms in which the digit 9 appears anywhere in the denominator are removed can be shown to converge to the value 22.92067661926415034816.... In fact, when all the terms containing any particular string of digits (in any
base) are removed, the series converges.
abRice, Adrian (2011). "The harmonic series: A primer". In Jardine, Dick;
Shell-Gellasch, Amy (eds.). Mathematical Time Capsules: Historical Modules for the Mathematics Classroom. MAA Notes. Vol. 77. Washington, DC: Mathematical Association of America. pp. 269–276.
^Mengoli, Pietro (1650).
"Praefatio [Preface]". Novae quadraturae arithmeticae, seu De additione fractionum [New arithmetic quadrature (i.e., integration), or On the addition of fractions] (in Latin). Bologna: Giacomo Monti.
Mengoli's proof is by contradiction: Let denote the sum of the series. Group the terms of the series in triplets: . Since for , , then , which is impossible for any finite . Therefore, the series diverges.
^Bernoulli, Jacob (1689). Propositiones arithmeticae de seriebus infinitis earumque summa finita [Arithmetical propositions about infinite series and their finite sums]. Basel: J. Conrad.
Changing the order of summation in the corresponding double series gives, in modern notation
abKnuth, Donald E. (1968). "1.2.7 Harmonic numbers".
The Art of Computer Programming, Volume I: Fundamental Algorithms (1st ed.). Addison-Wesley. pp. 73–78. Knuth writes, of the partial sums of the harmonic series "This sum does not occur very frequently in classical mathematics, and there is no standard notation for it; but in the analysis of algorithms it pops up nearly every time we turn around, and we will consistently use the symbol ... The letter stands for "harmonic", and we call a "harmonic number" because [the infinite series] is customarily called the harmonic series."
^Roy, Ranjan (December 2007). "Review of A Radical Approach to Real Analysis by David M. Bressoud". SIAM Review. 49 (4): 717–719.
JSTOR20454048. One might point out that Cauchy's condensation test is merely the extension of Oresme's argument for the divergence of the harmonic series