Harmonic series (mathematics)

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In mathematics, the harmonic series is the divergent infinite series:

\sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots

Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.

History

The fact that the harmonic series diverges was first proven in the 14th century by Nicole Oresme,^[1] but this achievement fell into obscurity. Proofs were given in the 17th century by Pietro Mengoli,^[2] Johann Bernoulli,^[3] and Jacob Bernoulli.^[4]^[5]

Historically, harmonic sequences have 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.^[6]

Paradoxes

The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the $n$ th term as $n$ goes to infinity is zero. The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the "worm on the rubber band".^[7] Suppose that a worm crawls along an infinitely-elastic one-meter rubber band at the same time as the rubber band is uniformly stretched. If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is "yes", for after $n$ minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is

{\frac {1}{100}}\sum _{k=1}^{n}{\frac {1}{k}}

(In fact the actual ratio is a little less than this sum as the band expands continuously.) The reason is that the band expands behind the worm also; eventually, the worm gets past the midway mark and the band behind expands increasingly more rapidly than the band in front.

Because the series gets arbitrarily large as $n$ becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. However, the value of $n$ at which this occurs must be extremely large: approximately $e 100$ , a number exceeding 10⁴³ minutes (10³⁷ years). Although the harmonic series does diverge, it does so very slowly.

Another problem involving the harmonic series is the Jeep problem.

Another example is the block-stacking problem: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling. The counterintuitive result is that one can stack them in such a way as to make the overhang arbitrarily large, provided there are enough dominoes.^[7]^[8]

A simpler example, on the other hand, is the swimmer that keeps adding more speed when touching the walls of the pool. The swimmer starts crossing a 10-meter pool at a speed of 2 m/s, and with every cross, another 2 m/s is added to the speed. In theory, the swimmer's speed is unlimited, but the number of pool crosses needed to get to that speed becomes very large; for instance, to get to the speed of light (ignoring special relativity), the swimmer needs to cross the pool 150 million times. Contrary to this large number, the time required to reach a given speed depends on the sum of the series at any given number of pool crosses (iterations):

{\frac {10}{2}}\sum _{k=1}^{n}{\frac {1}{k}}.

Calculating the sum (iteratively) shows that to get to the speed of light the time required is only 94 seconds. By continuing beyond this point (exceeding the speed of light, again ignoring special relativity), the time taken to cross the pool will in fact approach zero as the number of iterations becomes very large, and although the time required to cross the pool appears to tend to zero (at an infinite number of iterations), the sum of iterations (time taken for total pool crosses) will still diverge at a very slow rate.

Divergence

There are several well-known proofs of the divergence of the harmonic series. A few of them are given 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:

{\begin{aligned}&{}1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}+{\frac {1}{9}}+\cdots \\[12pt]>{}&1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots \end{aligned}}

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series. However, the sum of the second series is infinite:

{\begin{aligned}&{}1+\left({\frac {1}{2}}\right)+\left({\frac {1}{4}}+{\frac {1}{4}}\right)+\left({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}\right)+\left({\frac {1}{16}}+\cdots +{\frac {1}{16}}\right)+\cdots \\[12pt]={}&1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+\cdots =\infty \end{aligned}}

It follows (by the comparison test) that the sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that

\sum _{n=1}^{2^{k}}{\frac {1}{n}}\geq 1+{\frac {k}{2}}

for every positive integer $k$ .

This proof, proposed by Nicole Oresme in around 1350, is considered by many in the mathematical community to be a high point of medieval mathematics. It is still a standard proof taught in mathematics classes today. Cauchy's 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 $1 / n$ units high, so the total area of the infinite number of rectangles is the sum of the harmonic series:

{\begin{array}{c}{\text{area of}}\\{\text{rectangles}}\end{array}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots

Additionally, the total area under the curve $y = 1 / x$ from 1 to infinity is given by a divergent improper integral:

{\begin{array}{c}{\text{area under}}\\{\text{curve}}\end{array}}=\int _{1}^{\infty }{\frac {1}{x}}\,dx=\infty .

Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, this proves that

\sum _{n=1}^{k}{\frac {1}{n}}>\int _{1}^{k+1}{\frac {1}{x}}\,dx=\ln(k+1).

The generalization of this argument is known as the integral test.

Rate of divergence

The harmonic series diverges very slowly. For example, the sum of the first 10⁴³ terms is less than 100.^[9] This is because the partial sums of the series have logarithmic growth. In particular,

\sum _{n=1}^{k}{\frac {1}{n}}=\ln k+\gamma +\varepsilon _{k}\leq (\ln k)+1

where $γ$ is the Euler–Mascheroni constant and $ε k ~ 1 / 2 k$ which approaches 0 as $k$ goes to infinity. Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.

\sum _{p{\text{ prime }}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty .

Partial sums

The first thirty harmonic numbers
$n$	Partial sum of the harmonic series, $H n$
$n$	expressed as a fraction		decimal	relative size
1	1		~1	1
2	3	/2	~1.5	1.5
3	11	/6	~1.83333	1.83333
4	25	/12	~2.08333	2.08333
5	137	/60	~2.28333	2.28333
6	49	/20	~2.45	2.45
7	363	/140	~2.59286	2.59286
8	761	/280	~2.71786	2.71786
9	7003712900000000000♠7129	/7003252000000000000♠2520	~2.82897	2.82897
10	7003738100000000000♠7381	/7003252000000000000♠2520	~2.92897	2.92897
11	7004837110000000000♠83711	/7004277200000000000♠27720	~3.01988	3.01988
12	7004860210000000000♠86021	/7004277200000000000♠27720	~3.10321	3.10321
13	7006114599300000000♠1145993	/7005360360000000000♠360360	~3.18013	3.18013
14	7006117173300000000♠1171733	/7005360360000000000♠360360	~3.25156	3.25156
15	7006119575700000000♠1195757	/7005360360000000000♠360360	~3.31823	3.31823
16	7006243655900000000♠2436559	/7005720720000000000♠720720	~3.38073	3.38073
17	7007421422230000000♠42142223	/7007122522400000000♠12252240	~3.43955	3.43955
18	7007142743010000000♠14274301	/7006408408000000000♠4084080	~3.49511	3.49511
19	7008275295799000000♠275295799	/7007775975200000000♠77597520	~3.54774	3.54774
20	7007558351350000000♠55835135	/7007155195040000000♠15519504	~3.59774	3.59774
21	7007188580530000000♠18858053	/7006517316800000000♠5173168	~3.64536	3.64536
22	7007190931970000000♠19093197	/7006517316800000000♠5173168	~3.69081	3.69081
23	7008444316699000000♠444316699	/7008118982864000000♠118982864	~3.73429	3.73429
24	7009134782295500000♠1347822955	/7008356948592000000♠356948592	~3.77596	3.77596
25	7010340525224670000♠34052522467	/7009892371480000000♠8923714800	~3.81596	3.81596
26	7010343957422670000♠34395742267	/7009892371480000000♠8923714800	~3.85442	3.85442
27	7011312536252003000♠312536252003	/7010803134332000000♠80313433200	~3.89146	3.89146
28	7011315404588903000♠315404588903	/7010803134332000000♠80313433200	~3.92717	3.92717
29	7012922704651138700♠9227046511387	/7012232908956280000♠2329089562800	~3.96165	3.96165
30	7012930468283014700♠9304682830147	/7012232908956280000♠2329089562800	~3.99499	3.99499

The $n$ th partial sum of the diverging harmonic series,

H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},

is called the $n$ th harmonic number.

The difference between $H n$ and $ln n$ converges to the Euler–Mascheroni constant. The difference between any two harmonic numbers is never an integer. No harmonic numbers are integers, except for $H 1 = 1$ .^[10]

Related series

Alternating harmonic series

The first fourteen partial sums of the alternating harmonic series (black line segments) shown converging to the natural logarithm of 2 (red line).

The series

\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots

is known as the alternating harmonic series. This series converges by the alternating series test. In particular, the sum is equal to the natural logarithm of 2:

1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.

The alternating harmonic series, while conditionally convergent, is not absolutely convergent: if the terms in the series are systematically rearranged, in general the sum becomes different and, dependent on the rearrangement, possibly even infinite.

The alternating harmonic series formula is a special case of the Mercator series, the Taylor series for the natural logarithm.

A related series can be derived from the Taylor series for the arctangent:

\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots ={\frac {\pi }{4}}.

This is known as the Leibniz series.

General harmonic series

The general harmonic series is of the form

\sum _{n=0}^{\infty }{\frac {1}{an+b}},

where $a \neq 0$ and $b$ are real numbers and $b / a$ is not a non-positive integer.

By the limit comparison test with the harmonic series, all general harmonic series also diverge.

$p$ -series

A generalization of the harmonic series is the $p$ -series (or hyperharmonic series), defined as

\sum _{n=1}^{\infty }{\frac {1}{n^{p}}}

for any positive real number $p$ . When $p = 1$ , the $p$ -series is the harmonic series, which diverges. Either the integral test or the Cauchy condensation test shows that the $p$ -series converges for all $p > 1$ (in which case it is called the over-harmonic series) and diverges for all $p \leq 1$ . If $p > 1$ then the sum of the $p$ -series is $ζ (p)$ , i.e., the Riemann zeta function evaluated at $p$ .

The problem of finding the sum for $p = 2$ is called the Basel problem; Leonhard Euler showed it is $π$ ²/6. The value of the sum for $p = 3$ is called Apéry's constant.

ln-series

Related to the $p$ -series is the ln-series, defined as

\sum _{n=2}^{\infty }{\frac {1}{n(\ln n)^{p}}}

for any positive real number $p$ . This can be shown by the integral test to diverge for $p \leq 1$ but converge for all $p > 1$ .

$φ$ -series

For any convex, real-valued function $φ$ such that

\limsup _{u\to 0^{+}}{\frac {\varphi \left({\frac {u}{2}}\right)}{\varphi (u)}}<{\frac {1}{2}}

the series $\sum \infty n = 1 φ (1 / n)$ is convergent.

Random harmonic series

The random harmonic series

\sum _{n=1}^{\infty }{\frac {s_{n}}{n}},

where the $s n$ are independent, identically distributed random variables taking the values +1 and −1 with equal probability 1/2, is a well-known example in probability theory for a series of random variables that converges with probability 1. The fact of this convergence is an easy consequence of either the Kolmogorov three-series theorem or of the closely related Kolmogorov maximal inequality. Byron Schmuland of the University of Alberta further examined^[11] the properties of the random harmonic series, and showed that the convergent is a random variable with some interesting properties. In particular, the probability density function of this random variable evaluated at +2 or at −2 takes on the value 6999125000000000000♠0.124999999999999999999999999999999999999999764…, differing from 1/8 by less than 10⁻⁴². Schmuland's paper explains why this probability is so close to, but not exactly, 1/8. The exact value of this probability is given by the infinite cosine product integral $C 2$ ^[12] divided by $π$ .

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 and its value is less than 80.^[13] In fact, when all the terms containing any particular string of digits (in any base) are removed the series converges.

References

↑ Oresme, Nicole (c. 1360). Quaestiones super Geometriam Euclidis [Questions concerning Euclid's Geometry].
↑ Mengoli, Pietro (1650). "Praefatio [Preface]". Novae quadraturae arithmeticae, seu De additione fractionum [New arithmetic quadrature (i.e., integration), or On the addition of fractions]. Bologna: Giacomo Monti.
Mengoli's proof is by contradiction:
Let $S$ denote the sum of the series. Group the terms of the series in triplets: $S = 1 + (1 / 2 + 1 / 3 + 1 / 4) + (1 / 5 + 1 / 6 + 1 / 7) + (1 / 8 + 1 / 9 + 1 / 10) + \dots$ Since for $x > 1$ , $1 / x - 1 + 1 / x + 1 / x + 1 > 3 / x$ , then $S > 1 + 3 / 3 + 3 / 6 + 3 / 9 + \dots = 1 + 1 + 1 / 2 + 1 / 3 + \dots = 1 + S$ , which is false for any finite $S$ . Therefore, the series diverges.
↑ Bernoulli, Johann (1742). "Corollary III of De seriebus varia". Opera Omnia. Lausanne & Basel: Marc-Michel Bousquet & Co. vol. 4, p. 8.
↑ Bernoulli, Jacob (1689). Propositiones arithmeticae de seriebus infinitis earumque summa finita [Arithmetical propositions about infinite series and their finite sums]. Basel: J. Conrad.
↑ Bernoulli, Jacob (1713). Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis [Theory of inference, posthumous work. With the Treatise on infinite series…]. Basel: Thurneysen. pp. 250–251.
From p. 250, prop. 16:
"XVI. Summa serei infinita harmonicè progressionalium, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 &c. est infinita. Id primus deprehendit Frater:…"

[16. The sum of an infinite series of harmonic progression, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + …, is infinite. My brother first discovered this…]
↑ Hersey, George L. Architecture and Geometry in the Age of the Baroque. pp. 11–12, 37–51.
1 2 Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, pp. 258–264, ISBN 978-0-201-55802-9
↑ Sharp, R. T. (1954). "Problem 52: Overhanging dominoes" (PDF). Pi Mu Epsilon Journal. 1 (10): 411–412.
↑ "Sloane's A082912 : Sum of a(n) terms of harmonic series is > 10ⁿ". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ Weisstein, Eric Wolfgang. "Harmonic Number". MathWorld. Retrieved 2014-11-20.
↑ Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110: 407–416.
↑ Weisstein, Eric Wolfgang. "Infinite Cosine Product Integral". MathWorld. Retrieved 2010-11-14.
↑ "Nick's Mathematical Puzzles: Solution 72".

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Harmonic series", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
"The Harmonic Series Diverges Again and Again" (PDF). The AMATYC Review. 27: 31–43. 2006.
Weisstein, Eric Wolfgang. "Harmonic Series". MathWorld.

Weisstein, Eric Wolfgang. "Book Stacking Problem". MathWorld.

Hudelson, Matt (1 October 2010). "Proof Without Words: The Alternating Harmonic Series Sums to ln 2" (PDF). Mathematics Magazine. 83 (4): 294. doi:10.4169/002557010X521831.

Sequences and series

Integer
sequences

Basic	Arithmetic progression Geometric progression Square number Cubic number Factorial Powers of two Powers of 10
Advanced	Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number Full list

Properties of sequences

Properties of series

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/1^s + 1/2^s+ 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric
series

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