HOME
The Info List - Harmonic Series (mathematics)


--- Advertisement ---



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

n = 1

1 n

= 1 +

1 2

+

1 3

+

1 4

+

1 5

+ ⋯

displaystyle 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.

Contents

1 History 2 Paradoxes 3 Divergence

3.1 Comparison test 3.2 Integral
Integral
test

4 Rate of divergence 5 Partial sums 6 Related series

6.1 Alternating harmonic series 6.2 General harmonic series 6.3 p-series 6.4 ln-series 6.5 φ-series 6.6 Random harmonic series 6.7 Depleted harmonic series

7 See also 8 References 9 External links

History[edit] 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
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[edit] The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the nth 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

1 100

k = 1

n

1 k

displaystyle 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 also expands behind the worm; 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 e100, 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.

The block-stacking problem: blocks aligned according to the harmonic series bridges cleavages of any width.

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):

10 2

k = 1

n

1 k

.

displaystyle 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[edit] There are several well-known proofs of the divergence of the harmonic series. A few of them are given below. Comparison test[edit] 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:

1 +

1 2

+

1 3

+

1 4

+

1 5

+

1 6

+

1 7

+

1 8

+

1 9

+ ⋯

>

1 +

1 2

+

1

4

+

1 4

+

1

8

+

1

8

+

1

8

+

1 8

+

1

16

+ ⋯

displaystyle 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 color red mathbf 4 + frac 1 4 + frac 1 color red mathbf 8 + frac 1 color red mathbf 8 + frac 1 color red mathbf 8 + frac 1 8 + frac 1 color red mathbf 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:

1 +

(

1 2

)

+

(

1 4

+

1 4

)

+

(

1 8

+

1 8

+

1 8

+

1 8

)

+

(

1 16

+

+

1 16

)

+ ⋯

=

1 +

1 2

+

1 2

+

1 2

+

1 2

+ ⋯ = ∞

displaystyle 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

n = 1

2

k

1 n

≥ 1 +

k 2

displaystyle sum _ n=1 ^ 2^ k frac 1 n geq 1+ frac k 2

for every positive integer k. This proof, proposed by Nicole Oresme
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
Integral
test[edit]

Illustration of the 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:

area of

rectangles

= 1 +

1 2

+

1 3

+

1 4

+

1 5

+ ⋯

displaystyle 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:

area under

curve

=

1

1 x

d x = ∞ .

displaystyle 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

n = 1

k

1 n

>

1

k + 1

1 x

d x = ln ⁡ ( k + 1 ) .

displaystyle 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[edit] 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,

n = 1

k

1 n

= ln ⁡ k + γ +

ε

k

≤ ( ln ⁡ k ) + 1

displaystyle sum _ n=1 ^ k frac 1 n =ln k+gamma +varepsilon _ k leq (ln k)+1

where γ is the Euler–Mascheroni constant
Euler–Mascheroni constant
and εk ~ 1/2k which approaches 0 as k goes to infinity. Leonhard Euler
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.

p

 prime 

1 p

=

1 2

+

1 3

+

1 5

+

1 7

+

1 11

+

1 13

+

1 17

+ ⋯ = ∞ .

displaystyle 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[edit] Main article: Harmonic number

The first thirty harmonic numbers

n Partial sum of the harmonic series, Hn

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 finite partial sums of the diverging harmonic series,

H

n

=

k = 1

n

1 k

,

displaystyle H_ n =sum _ k=1 ^ n frac 1 k ,

are called harmonic numbers. The difference between Hn 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 H1 = 1.[10]:p. 24[11]:Thm. 1 Related series[edit] Alternating harmonic series[edit] See also: Riemann series theorem § Changing the sum

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

n = 1

( − 1

)

n + 1

n

= 1 −

1 2

+

1 3

1 4

+

1 5

− ⋯

displaystyle 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 −

1 2

+

1 3

1 4

+

1 5

− ⋯ = ln ⁡ 2.

displaystyle 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
Taylor series
for the natural logarithm. A related series can be derived from the Taylor series
Taylor series
for the arctangent:

n = 0

( − 1

)

n

2 n + 1

= 1 −

1 3

+

1 5

1 7

+ ⋯ =

π 4

.

displaystyle 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[edit] The general harmonic series is of the form

n = 0

1

a n + b

,

displaystyle sum _ n=0 ^ infty frac 1 an+b ,

where a ≠ 0 and b are real numbers and b/a is not a nonpositive integer. By the limit comparison test with the harmonic series, all general harmonic series also diverge.

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

n = 1

1

n

p

displaystyle 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 ≤ 1. If p > 1 then the sum of the p-series is ζ(p), i.e., the Riemann zeta function
Riemann zeta function
evaluated at p. The problem of finding the sum for p = 2 is called the Basel problem; Leonhard Euler
Leonhard Euler
showed it is π2/6. The value of the sum for p = 3 is called Apéry's constant. ln-series[edit] Related to the p-series is the ln-series, defined as

n = 2

1

n ( ln ⁡ n

)

p

displaystyle 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 ≤ 1 but converge for all p > 1. φ-series[edit] For any convex, real-valued function φ such that

lim sup

u →

0

+

φ ( u

/

2 )

φ ( u )

<

1 2

,

displaystyle limsup _ uto 0^ + frac varphi (u/2) varphi (u) < frac 1 2 ,

the series

n = 1

φ ( 1

/

n )

displaystyle textstyle sum _ n=1 ^ infty varphi (1/n)

is convergent.[citation needed] Random harmonic series[edit] The random harmonic series

n = 1

s

n

n

,

displaystyle sum _ n=1 ^ infty frac s_ n n ,

where the sn 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[12] 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 C2[13] divided by π. Depleted harmonic series[edit] Main article: Kempner 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.[14] In fact, when all the terms containing any particular string of digits (in any base) are removed the series converges. See also[edit]

Wikimedia Commons has media related to Harmonic series.

Harmonic progression List of sums of reciprocals

References[edit]

^ 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) + … Since for x > 1, 1/x − 1 + 1/x + 1/x + 1 > 3/x, then S > 1 + 3/3 + 3/6 + 3/9 + … = 1 + 1 + 1/2 + 1/3 + … = 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.  ^ a b Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989), Concrete Mathematics
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, N.J.A. (ed.). " Sequence
Sequence
A082912 (Sum of a(n) terms of harmonic series is > 10n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.  ^ Julian Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2009. ^ Thomas J. Osler, “Partial sums of series that cannot be an integer”, The Mathematical Gazette 96, November 2012, 515–519. https://www.jstor.org/stable/24496876?seq=1#page_scan_tab_contents ^ Schmuland, Byron (May 2003). "Random Harmonic Series" (PDF). American Mathematical Monthly. 110: 407–416. doi:10.2307/3647827.  ^ Weisstein, Eric W. "Infinite Cosine Product Integral". MathWorld. Retrieved 2010-11-14.  ^ "Nick's Mathematical Puzzles: Solution 72". 

External links[edit]

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

v t e

Sequences and series

Integer sequences

Basic

Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of 10

Advanced (list)

Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number

Properties of sequences

Cauchy sequence Monotone sequence Periodic sequence

Properties of series

Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping 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 + 1/1s + 1/2s+ 1/3s + ... (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

Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series

Hypergeometric series

Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hyp

.