Harmonic series (mathematics)
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: formula_1 The first formula_2 terms of the series sum to approximately formula_3, where formula_4 is the natural logarithm and formula_5 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the  sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the connected components of random graphs, the block-stacking problem on how far over the edge of a table a stack of blocks can be cantilevered, and the average case analysis of the quicksort algorithm. History. 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 proportions of floor plans, of elevations, and to establish harmonic relationships between both interior and exterior  architectural details of churches and palaces. The divergence of the harmonic series was first proven in 1350 by Nicole Oresme. Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. However, this achievement fell into obscurity. Additional proofs were published in the 17th century by Pietro Mengoli and by Jacob Bernoulli. Bernoulli credited his brother Johann Bernoulli for finding the proof, and it was later included in Johann Bernoulli's collected works. The partial sums of the harmonic series were named harmonic  numbers, and given their usual notation formula_6, in 1968 by Donald Knuth. Definition and divergence. The harmonic series is the infinite series formula_7 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. Comparison test. 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: formula_8 Grouping equal terms shows that the second series diverges: formula_9 Because each term of the harmonic series is greater than or equal to the corresponding term of the second series, it follows (by the comparison test) that  the harmonic series diverges as well. The same argument proves more strongly that, for every positive formula_10 This is the original proof given by Nicole Oresme in around 1350. The Cauchy condensation test is a generalization of this argument. Integral test. 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 formula_11 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 formula_12 stays entirely below the upper boundary of the rectangles, so the area under the curve (in the range of formula_13 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 improper integral, formula_14 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: formula_15 Generalizing this argument, any infinite sum of values of a monotone decreasing positive function (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. Partial sums. Adding the first formula_2 terms of the harmonic series produces a partial sum, called a harmonic number and formula_17 Growth rate. These numbers grow very slowly, with logarithmic growth, as can be seen from the integral test. More precisely, formula_18 where formula_19 is the Euler–Mascheroni constant and formula_20 which approaches 0 as formula_2 goes to infinity. Divisibility. No harmonic numbers are integers, except for One way to prove that formula_6 is not an integer is to consider the highest power of two formula_23 in the  range from If formula_24 is the least common multiple of the numbers from then formula_25 can be rewritten as a sum of fractions with equal denominators formula_26 in which only one of the numerators, is odd and the rest are even, and formula_24 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 formula_6 must be divisible by all prime numbers greater than formula_29, and uses Bertrand's postulate to prove that this set of primes is non-empty. The same argument implies more strongly that, except for formula_30, formula_31, and formula_32, 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. Interpolation. The digamma function is defined as  the logarithmic derivative of the gamma function formula_33 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 Applications. Many well-known mathematical problems have solutions involving the harmonic series and its partial sums. Crossing a desert. The 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 formula_2 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 formula_35 from the starting point and each other, where formula_36 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 formula_2th trip is formula_38 where formula_6 is the 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, formula_40: 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 . The problem has formula_41: 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 formula_42 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. Stacking blocks. In the block -stacking problem, one must place a pile of formula_2 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 formula_44 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 formula_45 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 formula_46 of its length extending beyond the next lower block, and so on. In this way, it is possible to place the formula_2 blocks in such a way that they extend formula_48 lengths beyond the table, where formula_6 is the 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. Counting primes and  divisors. In 1737, Leonhard Euler observed that, as a formal sum, the harmonic series is equal to an Euler product in which each term comes from a prime number: formula_50 where formula_51 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 formula_52 Here, each logarithm  is replaced by its Taylor series, and the constant formula_53 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. Another problem in number theory closely related to the harmonic series concerns the average number of divisors of the numbers in a range from 1 to formula_2, formalized as the average order of the divisor function, formula_55 The operation of  rounding each term in the harmonic series to the next smaller integer multiple of formula_11 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 formula_57 (expressed in big O notation). Bounding the final error term more precisely remains an open problem, known as Dirichlet's divisor problem. Collecting coupons. 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 connectivity of random graphs. In situations of this form, once there are formula_58 items remaining to be collected out of a total of formula_2 equally-likely items, the probability of collecting a new item in a single random choice is formula_60 and the expected number of random choices needed until a new item is collected  Summing over all values of formula_58 from formula_2 shows that the total expected number of random choices needed to collect all items where formula_6 is the harmonic number. Analyzing algorithms. The 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 formula_13 and formula_65 are separated by formula_58 other items, then the algorithm will make a comparison between formula_13 and formula_65 only when, as the recursion progresses, it picks formula_13 or formula_65 as a pivot before  picking any of the other formula_58 items between them. Because each of these formula_72 items is equally likely to be chosen first, this happens with probability formula_73. 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 formula_74 The divergence of the harmonic series corresponds in this application to the fact that, in the comparison model of sorting used for quicksort, it is not possible to sort in linear time. Related series. Alternating harmonic series. The series formula_75 is known as the alternating  harmonic series. It is conditionally convergent by the alternating series test, but not absolutely convergent. Its sum is the natural logarithm of 2. Using alternating signs with only odd unit fractions produces a related series, the Leibniz formula for formula_76 Riemann zeta function. The Riemann zeta function is defined for real formula_77 by the convergent series formula_78 which for formula_79 would be the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers where the extended function has a simple pole. Other important values of the zeta function include the solution to the  Basel problem, Apéry's constant proved by Roger Apéry to be an irrational number, and the "critical line" of complex numbers with conjectured by the Riemann hypothesis to be the only values other than negative integers where the function can be zero. Random harmonic series. The random harmonic series is formula_80 where the values formula_81 are independent and identically distributed random variables that take the two values formula_82 and formula_83 with equal It converges with probability 1, as can be seen by using the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. The sum of the  series is a random variable whose probability density function is for values between and decreases to near-zero for values greater or less Intermediate between these ranges, at the the probability density is formula_84 for a nonzero but very small value Depleted harmonic series. 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 . In fact, when all the terms containing any particular string of digits (in any base) are removed, the series converges.