View source for Harmonic series (mathematics)

{{Calculus |Series}}

In [[mathematics]], the '''harmonic series''' is the [[Divergent series|divergent]] [[infinite series]]:

: <math>\sum_{n=1}^\infty\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots</math>

Its name derives from the concept of [[overtone]]s, or harmonics [[harmonic series (music)|in music]]: the [[wavelength]]s of the overtones of a vibrating string are {{sfrac|1|2}}, {{sfrac|1|3}}, {{sfrac|1|4}}, etc., of the string's [[Fundamental frequency|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]],<ref>{{cite book|authorlink=Nicole Oresme|first=Nicole|last=Oresme|date=c. 1360|title=Quaestiones super Geometriam Euclidis|trans-title=Questions concerning Euclid's Geometry}}</ref> but this achievement fell into obscurity. Proofs were given in the 17th century by [[Pietro Mengoli]],<ref>{{cite book|authorlink=Pietro Mengoli|first=Pietro|last=Mengoli|title=Novae quadraturae arithmeticae, seu De additione fractionum|trans-title=New arithmetic quadrature (i.e., integration), or On the addition of fractions|location=Bologna|publisher=Giacomo Monti|date=1650|chapter-url=https://books.google.com/books?id=f9eM5uQvRucC&pg=PP9#v=onepage&q&f=false|chapter=Praefatio [Preface]}}<br>
Mengoli's proof is by contradiction:
:Let {{mvar|S}} denote the sum of the series.  Group the terms of the series in triplets: {{math|''S'' {{=}} 1 + ({{sfrac|1|2}} + {{sfrac|1|3}} + {{sfrac|1|4}}) + ({{sfrac|1|5}} + {{sfrac|1|6}} + {{sfrac|1|7}}) + ({{sfrac|1|8}} + {{sfrac|1|9}} + {{sfrac|1|10}}) + …}} Since for {{math|''x'' > 1}}, {{math|{{sfrac|1|''x'' − 1}} + {{sfrac|1|''x''}} + {{sfrac|1|''x'' + 1}} > {{sfrac|3|''x''}}}}, then {{math|''S'' > 1 + {{sfrac|3|3}} + {{sfrac|3|6}} + {{sfrac|3|9}} + … {{=}} 1 + 1 + {{sfrac|1|2}} + {{sfrac|1|3}} + … {{=}} 1 + ''S''}}, which is false for any finite {{mvar|S}}.  Therefore, the series diverges.</ref> [[Johann Bernoulli]],<ref>{{cite book|authorlink=Johann Bernoulli|chapter=Corollary III of ''De seriebus varia''|first=Johann|last=Bernoulli|title=Opera Omnia|location=Lausanne & Basel|publisher=Marc-Michel Bousquet & Co.|date=1742|at=vol. 4, p. 8|chapter-url=https://books.google.com/books?id=sxUOAAAAQAAJ&pg=PA6#v=onepage&q&f=false}}</ref> and [[Jacob Bernoulli]].<ref>{{cite book|authorlink=Jacob Bernoulli|first=Jacob|last=Bernoulli|title=Propositiones arithmeticae de seriebus infinitis earumque summa finita|trans-title=Arithmetical propositions about infinite series and their finite sums|location=Basel|publisher=J. Conrad|date=1689}}</ref><ref>{{cite book|authorlink=Jacob Bernoulli|first=Jacob|last1=Bernoulli|title=Ars conjectandi, opus posthumum.  Accedit Tractatus de seriebus infinitis|trans-title=Theory of inference, posthumous work. With the Treatise on infinite series…|location=Basel|publisher=Thurneysen|date=1713|url=https://books.google.com/books?id=CF4UAAAAQAAJ&pg=PA250#v=onepage&q&f=false|pages=250–251}}<br>From p. 250, prop. 16:
:"''XVI. Summa serei infinita harmonicè progressionalium, {{sfrac|1|1}} + {{sfrac|1|2}} + {{sfrac|1|3}} + {{sfrac|1|4}} + {{sfrac|1|5}} &c. est infinita. Id primus deprehendit Frater:…''"
:[16.  The sum of an infinite series of harmonic progression, {{sfrac|1|1}} + {{sfrac|1|2}} + {{sfrac|1|3}} + {{sfrac|1|4}} + {{sfrac|1|5}} + …, is infinite. My brother first discovered this…]</ref>

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 [[Proportion (architecture)|proportions]] of [[Architectural drawing#Floor plan|floor plans]], of [[Architectural drawing#Elevation|elevations]], and to establish harmonic relationships between both interior and exterior architectural details of churches and palaces.<ref>{{cite book|first=George L.|last=Hersey|title=Architecture and Geometry in the Age of the Baroque|pages=11–12, 37–51}}</ref>

==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]]:

:<math>\begin{align}
&{} 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]
\ge {} &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{align}</math>

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 or equal to the sum of the second series.  However, the sum of the second series is infinite:

:<math>\begin{align}
&{} 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{align}</math>

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

:<math>\sum_{n=1}^{2^k} \frac{1}{n} \geq 1 + \frac{k}{2}</math>

for every [[Positive number|positive]] [[integer]] {{mvar|k}}.

This proof, proposed by [[Nicole Oresme]] in around 1350, is considered by many in the mathematical community{{by whom|date=September 2018}} to be a high point of [[History of mathematics#Roman and medieval European mathematics|medieval mathematics]]. It is still a standard proof taught in mathematics classes today. [[Cauchy condensation test|Cauchy's condensation test]] is a generalization of this argument.

===Integral test===
[[Image:Integral Test.svg|thumb|right|250px|
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 {{math|{{sfrac|1|''n''}}}} units high, so the total area of the infinite number of rectangles is the sum of the harmonic series:

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

Additionally, the total area under the curve {{math|''y'' {{=}} {{sfrac|1|''x''}}}} from 1 to infinity is given by a divergent [[improper integral]]:

:<math>\begin{array}{c} \text{area under} \\ \text{curve}\end{array}
= \int_1^\infty\frac{1}{x}\,dx = \infty.</math>

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

:<math> \sum_{n=1}^k \frac{1}{n} > \int_1^{k+1} \frac{1}{x}\,dx = \ln(k+1). </math>

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 10<sup>43</sup> terms is less than 100.<ref>{{Cite OEIS|sequencenumber=A082912|name=Sum of ''a''(''n'') terms of harmonic series is > 10<sup>''n''</sup>}}</ref>  This is because the partial sums of the series have [[logarithmic growth]]. In particular,
:<math>\sum_{n=1}^k\frac{1}{n} = \ln k + \gamma + \varepsilon_k \leq (\ln k) + 1</math>
where {{mvar|γ}} is the [[Euler–Mascheroni constant]] and {{math|''ε<sub>k</sub>'' ~ {{sfrac|1|2''k''}}}} which approaches 0 as {{mvar|k}} goes to infinity. [[Leonhard Euler]] proved both this and also the more striking fact that the sum which includes only [[the sum of the reciprocals of the primes diverges|the reciprocals of primes]] also diverges, i.e.

:<math>\sum_{p\text{ prime }}\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \frac1{13} + \frac1{17} +\cdots = \infty.</math>

==Partial sums==
{{main|Harmonic number}}
{| class="wikitable mw-collapsible mw-collapsed infobox" style="font-size:100%;line-height:0.8;text-align:left;"
|+ {{small|{{nobreak|The first thirty harmonic numbers}}}}
! rowspan="2" style="padding-top:1em;"|{{mvar|n}} !! colspan="4"|Partial sum of the harmonic series, {{mvar|H<sub>n</sub>}}
|-
! colspan="2"|expressed as a fraction !! decimal !! relative size
|-
| style="text-align:right;"|1 || style="text-align:center;" colspan="2"|1 || {{0|~}}{{bartable|1||20}}
|-
| style="text-align:right;"|2 || style="border-right:none;text-align:right;padding-right:0;"|3 || style="border-left:none;padding-left:0;"|/2 || {{0|~}}{{bartable|1.5||20}}
|-
| style="text-align:right;"|3 || style="border-right:none;text-align:right;padding-right:0;"|11 || style="border-left:none;padding-left:0;"|/6 || ~{{bartable|1.83333||20}}
|-
| style="text-align:right;"|4 || style="border-right:none;text-align:right;padding-right:0;"|25 || style="border-left:none;padding-left:0;"|/12 || ~{{bartable|2.08333||20}}
|-
| style="text-align:right;"|5 || style="border-right:none;text-align:right;padding-right:0;"|137 || style="border-left:none;padding-left:0;"|/60 || ~{{bartable|2.28333||20}}
|-
| style="text-align:right;"|6 || style="border-right:none;text-align:right;padding-right:0;"|49 || style="border-left:none;padding-left:0;"|/20 || {{0|~}}{{bartable|2.45||20}}
|-
| style="text-align:right;"|7 || style="border-right:none;text-align:right;padding-right:0;"|363 || style="border-left:none;padding-left:0;"|/140 || ~{{bartable|2.59286||20}}
|-
| style="text-align:right;"|8 || style="border-right:none;text-align:right;padding-right:0;"|761 || style="border-left:none;padding-left:0;"|/280 || ~{{bartable|2.71786||20}}
|-
| style="text-align:right;"|9 || style="border-right:none;text-align:right;padding-right:0;"|{{val|7129|fmt=gaps}} || style="border-left:none;padding-left:0;"|/{{val|2520|fmt=gaps}} || ~{{bartable|2.82897||20}}
|-
| style="text-align:right;"|10 || style="border-right:none;text-align:right;padding-right:0;"|{{val|7381|fmt=gaps}} || style="border-left:none;padding-left:0;"|/{{val|2520|fmt=gaps}} || ~{{bartable|2.92897||20}}
|-
| style="text-align:right;"|11 || style="border-right:none;text-align:right;padding-right:0;"|{{val|83711}} || style="border-left:none;padding-left:0;"|/{{val|27720}} || ~{{bartable|3.01988||20}}
|-
| style="text-align:right;"|12 || style="border-right:none;text-align:right;padding-right:0;"|{{val|86021}} || style="border-left:none;padding-left:0;"|/{{val|27720}} || ~{{bartable|3.10321||20}}
|-
| style="text-align:right;"|13 || style="border-right:none;text-align:right;padding-right:0;"|{{val|1145993}} || style="border-left:none;padding-left:0;"|/{{val|360360}} || ~{{bartable|3.18013||20}}
|-
| style="text-align:right;"|14 || style="border-right:none;text-align:right;padding-right:0;"|{{val|1171733}} || style="border-left:none;padding-left:0;"|/{{val|360360}} || ~{{bartable|3.25156||20}}
|-
| style="text-align:right;"|15 || style="border-right:none;text-align:right;padding-right:0;"|{{val|1195757}} || style="border-left:none;padding-left:0;"|/{{val|360360}} || ~{{bartable|3.31823||20}}
|-
| style="text-align:right;"|16 || style="border-right:none;text-align:right;padding-right:0;"|{{val|2436559}} || style="border-left:none;padding-left:0;"|/{{val|720720}} || ~{{bartable|3.38073||20}}
|-
| style="text-align:right;"|17 || style="border-right:none;text-align:right;padding-right:0;"|{{val|42142223}} || style="border-left:none;padding-left:0;"|/{{val|12252240}} || ~{{bartable|3.43955||20}}
|-
| style="text-align:right;"|18 || style="border-right:none;text-align:right;padding-right:0;"|{{val|14274301}} || style="border-left:none;padding-left:0;"|/{{val|4084080}} || ~{{bartable|3.49511||20}}
|-
| style="text-align:right;"|19 || style="border-right:none;text-align:right;padding-right:0;"|{{val|275295799}} || style="border-left:none;padding-left:0;"|/{{val|77597520}} || ~{{bartable|3.54774||20}}
|-
| style="text-align:right;"|20 || style="border-right:none;text-align:right;padding-right:0;"|{{val|55835135}} || style="border-left:none;padding-left:0;"|/{{val|15519504}} || ~{{bartable|3.59774||20}}
|-
| style="text-align:right;"|21 || style="border-right:none;text-align:right;padding-right:0;"|{{val|18858053}} || style="border-left:none;padding-left:0;"|/{{val|5173168}} || ~{{bartable|3.64536||20}}
|-
| style="text-align:right;"|22 || style="border-right:none;text-align:right;padding-right:0;"|{{val|19093197}} || style="border-left:none;padding-left:0;"|/{{val|5173168}} || ~{{bartable|3.69081||20}}
|-
| style="text-align:right;"|23 || style="border-right:none;text-align:right;padding-right:0;"|{{val|444316699}} || style="border-left:none;padding-left:0;"|/{{val|118982864}} || ~{{bartable|3.73429||20}}
|-
| style="text-align:right;"|24 || style="border-right:none;text-align:right;padding-right:0;"|{{val|1347822955}} || style="border-left:none;padding-left:0;"|/{{val|356948592}} || ~{{bartable|3.77596||20}}
|-
| style="text-align:right;"|25 || style="border-right:none;text-align:right;padding-right:0;"|{{val|34052522467}} || style="border-left:none;padding-left:0;"|/{{val|8923714800}} || ~{{bartable|3.81596||20}}
|-
| style="text-align:right;"|26 || style="border-right:none;text-align:right;padding-right:0;"|{{val|34395742267}} || style="border-left:none;padding-left:0;"|/{{val|8923714800}} || ~{{bartable|3.85442||20}}
|-
| style="text-align:right;"|27 || style="border-right:none;text-align:right;padding-right:0;"|{{val|312536252003}} || style="border-left:none;padding-left:0;"|/{{val|80313433200}} || ~{{bartable|3.89146||20}}
|-
| style="text-align:right;"|28 || style="border-right:none;text-align:right;padding-right:0;"|{{val|315404588903}} || style="border-left:none;padding-left:0;"|/{{val|80313433200}} || ~{{bartable|3.92717||20}}
|-
| style="text-align:right;"|29 || style="border-right:none;text-align:right;padding-right:0;"|{{val|9227046511387}} || style="border-left:none;padding-left:0;"|/{{val|2329089562800}} || ~{{bartable|3.96165||20}}
|-
| style="text-align:right;"|30 || style="border-right:none;text-align:right;padding-right:0;"|{{val|9304682830147}} || style="border-left:none;padding-left:0;"|/{{val|2329089562800}} || ~{{bartable|3.99499||20}}
|}
<!-- Python script to generate n from 2 to 30:
import fractions
numerator = 1; denominator = 1
for i in range(2, 30 + 1):
 numerator = numerator * i + denominator; denominator *= i; gcd = fractions.gcd(numerator, denominator); numerator /= gcd; denominator /= gcd
 decimal   = ('{}' if (i < 3 or i == 6) else '{:.5f}').format(float(numerator) / denominator); exact = '' if (i < 3 or i == 6) else '~'
 print('|-\n| style="text-align:right;"|{} || style="border-right:none;text-align:right;padding-right:0;"|{:,} || style="border-left:none;padding-left:0;"|/{:,} || {}{{{{bartable|{}||20}}}}'.
       format(i, numerator, denominator, exact, decimal).replace(',', ''))
-->
The finite partial sums of the diverging harmonic series,

: <math>H_n = \sum_{k = 1}^n \frac{1}{k},</math>

are called [[harmonic number]]s.

The difference between {{mvar|H<sub>n</sub>}} and {{math|[[natural logarithm|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 {{math|''H''<sub>1</sub> {{=}} 1}}.<ref>Julian Havil, 
''Gamma: Exploring Euler’s Constant'', Princeton University Press, 2009.</ref>{{rp|p. 24}}<ref>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</ref>{{rp|Thm. 1}}

==Related series==

===Alternating harmonic series===
{{See also|Riemann series theorem#Changing the sum}}
[[Image:Alternating Harmonic Series.PNG|right|thumb|240px|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

: <math>\sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots</math>

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]]:

:<math>1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots = \ln 2.</math>

The alternating harmonic series, while [[conditional convergence|conditionally convergent]], is not [[absolute convergence|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]]:

: <math>\sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \frac{\pi}{4}.</math>

This is known as the [[Leibniz series]].

===General harmonic series===

The '''general harmonic series''' is of the form

:<math>\sum_{n=0}^{\infty}\frac{1}{an+b} ,</math>

where {{math|''a'' ≠ 0}} and {{mvar|b}} are real numbers, and {{mvar|{{sfrac|b|a}}}} is not zero or a negative integer.<!--The current wording is intentional and correct; please don't change this to "is a positive integer"-->

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

{{anchor|P-series}}

==={{mvar|p}}-series===

A generalization of the harmonic series is the '''{{mvar|p}}-series''' (or '''hyperharmonic series'''), defined as

:<math>\sum_{n=1}^{\infty}\frac{1}{n^p}</math>

for any real number {{mvar|p}}. When {{math|''p'' {{=}} 1}}, the {{mvar|p}}-series is the harmonic series, which diverges. Either the [[integral test]] or the [[Cauchy condensation test]] shows that the {{mvar|p}}-series converges for all {{math|''p'' > 1}} (in which case it is called the '''over-harmonic series''') and diverges for all {{math|''p'' ≤ 1}}. If {{math|''p'' > 1}} then the sum of the {{mvar|p}}-series is {{math|''ζ''(''p'')}}, i.e., the [[Riemann zeta function]] evaluated at {{mvar|p}}.

The problem of finding the sum for {{math|''p'' {{=}} 2}} is called the [[Basel problem]]; [[Leonhard Euler]] showed it is {{sfrac|{{pi}}<sup>2</sup>|6}}. The value of the sum for {{math|''p'' {{=}} 3}} is called [[Apéry's constant]], since [[Roger Apéry]] proved that it is an [[irrational number]].

===ln-series===

Related to the '''{{mvar|p}}-series''' is the '''ln-series''', defined as

:<math>\sum_{n=2}^{\infty}\frac{1}{n (\ln n)^p}</math>

for any positive real number {{mvar|p}}. This can be shown by the integral test to diverge for {{math|''p'' ≤ 1}} but converge for all {{math|''p'' > 1}}.

==={{mvar|φ}}-series===

For any [[Convex function|convex]], real-valued function {{mvar|φ}} such that

:<math>\limsup_{u\to 0^+}\frac{\varphi\left(\frac{u}{2}\right)}{\varphi(u)} < \frac{1}{2}, </math>

the series

:<math>\sum_{n=1}^\infty \varphi\left(\frac{1}{n}\right)</math>

is convergent.{{citation needed|date=February 2018}}

===Random harmonic series===

The random harmonic series

:<math>\sum_{n=1}^{\infty}\frac{s_{n}}{n},</math>

where the {{mvar|s<sub>n</sub>}} are [[statistical independence|independent]], identically distributed random variables taking the values +1 and −1 with equal probability {{sfrac|1|2}}, is a well-known example in probability theory for a series of random variables that converges [[Almost surely|with probability 1]]. The fact of this convergence is an easy consequence of either the [[Kolmogorov's three-series theorem|Kolmogorov three-series theorem]] or of the closely related [[Kolmogorov's inequality|Kolmogorov maximal inequality]]. Byron Schmuland of the University of Alberta further examined<ref>{{cite journal|first=Byron|last=Schmuland|title=Random Harmonic Series|journal=American Mathematical Monthly|volume=110|pages=407–416|date=May 2003|url=http://www.stat.ualberta.ca/people/schmu/preprints/rhs.pdf|doi=10.2307/3647827}}</ref> the properties of the random harmonic series, and showed that the convergent series 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 {{val|0.124999999999999999999999999999999999999999764}}..., differing from {{sfrac|1|8}} by less than 10<sup>−42</sup>. Schmuland's paper explains why this probability is so close to, but not exactly, {{sfrac|1|8}}.  The exact value of this probability is given by the infinite cosine product integral {{math|''C''<sub>2</sub>}}<ref>{{MathWorld|title=Infinite Cosine Product Integral|id=InfiniteCosineProductIntegral|accessdate=2010-11-14}}</ref> divided by {{pi}}.

===Depleted harmonic series===
{{main|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.<ref>{{cite web|url=http://www.qbyte.org/puzzles/p072s.html|title=Nick's Mathematical Puzzles: Solution 72}}</ref>  In fact, when all the terms containing any particular string of digits (in any [[number base|base]]) are removed the series converges.

==Applications==
The harmonic series can be [[counterintuitive]] to students first encountering it, because it is a [[divergent series]] even though the limit of the {{mvar|n}}th term as {{mvar|n}} goes to infinity is zero.  The divergence of the harmonic series is also the source of some apparent [[paradox]]es. One example of these is the "[[Ant on a rubber rope|worm on the rubber band]]".<ref name="autogenerated258">{{Citation | last1=Graham | first1=Ronald | author1-link=Ronald Graham | last2=Knuth | first2=Donald E. | author2-link=Donald Knuth | last3=Patashnik | first3=Oren | author3-link=Oren Patashnik | title=Concrete Mathematics | publisher=[[Addison-Wesley]] | edition=2nd | isbn=978-0-201-55802-9 | year=1989 | pages=258–264}}</ref> 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 {{mvar|n}} minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is

:<math>\frac{1}{100}\sum_{k=1}^n\frac{1}{k}.</math>

(In fact the actual ratio is a little less than this sum as the band expands continuously.)

Because the series gets arbitrarily large as {{mvar|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 {{mvar|n}} at which this occurs must be extremely large: approximately {{math|[[e (mathematical constant)|''e'']]<sup>100</sup>}}, a number exceeding 10<sup>43</sup> minutes (10<sup>37</sup> years). Although the harmonic series does diverge, it does so very slowly.

Another problem involving the harmonic series is the [[Jeep problem]], which (in one form) asks how much total fuel is required for a [[Willys MB|jeep]] with a limited fuel-carrying capacity to cross a desert, possibly leaving fuel drops along the route.  The distance that can be traversed with a given amount of fuel is related to the partial sums of the harmonic series, which grow logarithmically.  And so the fuel required increases exponentially with the desired distance.

[[File:Block_stacking_problem.svg|thumb|250px|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.<ref name="autogenerated258"/><ref>{{cite journal|first=R. T.|last=Sharp|title=Problem 52: Overhanging dominoes|journal = Pi Mu Epsilon Journal|year=1954|pages=411–412|volume=1|issue=10|url=http://www.pme-math.org/journal/issues/PMEJ.Vol.1.No.10.pdf}}</ref>

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&nbsp;m/s, and with every cross, another 2&nbsp;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&nbsp;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):

:<math>\frac{10}{2}\sum_{k=1}^n\frac{1}{k}.</math>

Calculating the sum (iteratively) shows that to get to the speed of light the time required is only 97&nbsp;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.

==See also==
{{Commons category|Harmonic series}}
* [[Harmonic progression (mathematics)|Harmonic progression]]
* [[List of sums of reciprocals]]

==References==
{{reflist|30em}}

==External links==
* {{springer|title=Harmonic series|id=p/h046540}}
<!-- * [http://eom.springer.de/H/h046540.htm "Harmonic Series" at Springer Encyclopaedia of Mathematics] -->
* {{cite journal|url=http://prairiestate.edu/skifowit/harmapa.pdf|title=The Harmonic Series Diverges Again and Again|journal=The AMATYC Review|volume=27|date=2006|pages=31–43}}
* {{MathWorld|title=Harmonic Series|urlname=HarmonicSeries}}
* {{MathWorld|title=Book Stacking Problem|urlname=BookStackingProblem}}
* {{cite journal|last1=Hudelson|first1=Matt|title=Proof Without Words: The Alternating Harmonic Series Sums to ln 2|journal=Mathematics Magazine|date=1 October 2010|volume=83|issue=4|page=294|doi=10.4169/002557010X521831|url=http://www.maa.org/sites/default/files/Hudleson-MMz-201007804.pdf}}

{{Series (mathematics)}}

[[Category:Divergent series]]
[[Category:Proof without words]]