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| {{Calculus |Series}}
| | Hello and welcome. My title is Numbers Wunder. One of the extremely very best things in the world for me is to do aerobics and I've been doing it for quite a while. Bookkeeping is my profession. Years in the past we moved to Puerto Rico and my family enjoys it.<br><br>My web page: [http://C.Judgementgaming.com/diettogorevi C.Judgementgaming.com] |
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| In [[mathematics]], the '''harmonic series''' is the [[Divergent series|divergent]] [[infinite series]]:
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| : <math>\sum_{n=1}^\infty\,\frac{1}{n} \;\;=\;\; 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.\!</math>
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| Its name derives from the concept of [[overtone]]s, or harmonics [[harmonic series (music)|in music]]: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 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.
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| ==History==
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| The fact that the harmonic series diverges was first proven in the 14th century by [[Nicole Oresme]],<ref>Nicole Oresme (ca. 1360) ''Quastiones super Geometriam Euclidis'' (Questions concerning Euclid's Geometry).</ref> but this achievement fell into obscurity. Proofs were given in the 17th century by [[Pietro Mengoli]],<ref>Pietro Mengoli, ''Novæ quadraturæ arithmeticæ, seu De additione fractionum'' [New arithmetic quadrature (i.e., integration), or On the addition of fractions] (Bologna ("Bononiæ"), (Italy): Giacomo Monti ("Jacobi Monti"), 1650). The proof of the divergence of the harmonic series is presented in the book's [http://books.google.com/books?id=f9eM5uQvRucC&pg=PP9#v=onepage&q&f=false preface (Præfatio)].<br>
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| Mengoli's proof is by contradiction:<br>
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| Let S denote the sum of the series. Group the terms of the series in triplets:<br>
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| S = 1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + (1/8 + 1/9 + 1/10) + …<br>
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| Since for x > 1, 1/(x-1) + 1/x + 1/(x+1) > 3/x, then<br>
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| 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.</ref> [[Johann Bernoulli]],<ref>See: Corollary III of ''De seriebus varia'' in: Johannis Bernoulli, ''Opera Omnia'' (Lausanne & Basel, Switzerland: Marc-Michel Bousquet & Co., 1742), vol. 4, [http://books.google.com/books?id=sxUOAAAAQAAJ&pg=PA6#v=onepage&q&f=false p. 8, Corollary III.]</ref> and [[Jacob Bernoulli]].<ref>See:
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| * Jacob Bernoulli, ''Propositiones arithmeticae de seriebus infinitis earumque summa finita'' [Arithmetical propositions about infinite series and their finite sums] (Basel, Switzerland: J. Conrad, 1689).
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| * Jacob Bernoulli, ''Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis'' … [Theory of inference, posthumous work. With the "Treatise on infinite series" joined … ] (Basel, Switzerland: Thurneysen, 1713), [http://books.google.com/books?id=CF4UAAAAQAAJ&pg=PA250#v=onepage&q&f=false pp. 250-251.] From page 250, proposition 16:<br>''"XVI. Summa serei infinita harmonicè progressionalium, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 &c. est infinita.''
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| :''Id primus deprehendit Frater: … "''<br>
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| (16. The sum of an infinite series of harmonic progression, 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + … , is infinite.<br>
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| [My] brother first discovered it [i.e., this proof].)</ref>
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| 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>George L. Hersey, ''Architecture and Geometry in the Age of the Baroque'', p 11-12 and p37-51.</ref>
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| ==Paradoxes==
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| The harmonic series is counterintuitive to students first encountering it, because it is a [[divergent series]] 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 [[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 a 1 metre rubber band and, after each minute, the rubber band is uniformly stretched by an additional 1 metre. If the worm travels 1 centimetre 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
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| :<math>\frac{1}{100}\sum_{k=1}^n\frac{1}{k}.</math>
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| 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. The value of ''n'' at which this occurs must be extremely large, however, approximately ''e''<sup>100</sup>, a number exceeding 10<sup>40</sup>. Although the harmonic series does diverge, it does so very slowly.
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| Another example is: 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>{{citation|first=R.T.|last=Sharp|title=Problem 52: Overhanging dominoes|journal = Pi Mu Epsilon Journal|year=1954|pages=411–412}}</ref>
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| ==Divergence==
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| There are several well-known proofs of the divergence of the harmonic series. Two of them are given below.
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| ===Comparison test===
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| One way to prove divergence is to compare the harmonic series with another divergent series:
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| : <math>
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| \begin{align}
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| & 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]
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| >\;\;\; & 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.
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| \end{align}
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| </math>
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| 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:
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| :<math>
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| \begin{align}
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| & 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]
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| =\;\; & 1 \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \frac{1}{2} \;\;+\;\; \cdots \;\;=\;\; \infty.
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| \end{align}
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| </math>
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| 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
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| :<math>\sum_{n=1}^{2^k} \,\frac{1}{n} \;\geq\; 1 + \frac{k}{2}</math>
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| for every [[Positive number|positive]] [[integer]] ''k''. | |
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| This proof, due to [[Nicole Oresme]], is considered by many in the mathematical community 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.
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| ===Integral test===
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| [[Image:Integral Test.svg|thumb|right|250px]]
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| 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 rectangles is the sum of the harmonic series:
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| :<math>
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| \begin{array}{c}
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| \text{area of}\\
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| \text{rectangles}
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| \end{array}
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| = 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots.
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| </math> | |
| However, the total area under the curve ''y'' = 1 / ''x'' from 1 to infinity is given by an [[improper integral]]:
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| :<math>
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| \begin{array}{c}
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| \text{area under}\\
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| \text{curve}
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| \end{array}
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| = \int_1^\infty\frac{1}{x}\,dx \;=\; \infty.
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| </math>
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| 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
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| :<math>
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| \sum_{n=1}^k \, \frac{1}{n} \;>\; \int_1^{k+1} \frac{1}{x}\,dx \;=\; \ln(k+1).
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| </math>
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| The generalization of this argument is known as the [[integral test]].
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| ==Rate of divergence==
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| The harmonic series diverges very slowly. For example, the sum of the first 10<sup>43</sup> terms is less than 100.<ref>{{SloanesRef |sequencenumber=A082912|name=Sum of a(n) terms of harmonic series is > 10^n}}</ref> This is because the partial sums of the series have [[logarithmic growth]]. In particular,
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| :<math>\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k < \ln k + 1</math>
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| where <math>\gamma</math> is the [[Euler–Mascheroni constant]] and <math>\varepsilon_k</math> ~ <math>\frac{1}{2k}</math> which approaches 0 as <math>k</math> 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.
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| :<math>\sum_{p\text{ prime }}\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \frac1{13} + \frac1{17} +\cdots = \infty.</math>
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| ==Partial sums==
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| The ''n''th partial sum of the diverging harmonic series,
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| : <math>H_n = \sum_{k = 1}^n \frac{1}{k},\!</math>
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| is called the ''n''th '''[[harmonic number]]'''.
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| The difference between the ''n''th harmonic number and the [[natural logarithm]] of ''n'' converges to the [[Euler–Mascheroni constant]].
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| The difference between distinct harmonic numbers is never an integer.
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| No harmonic numbers are integers, except for ''n'' = 1.<ref>http://mathworld.wolfram.com/HarmonicNumber.html</ref>
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| ==Related series==
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| ===Alternating harmonic series===
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| [[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).]]
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| The series
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| : <math>
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| \sum_{n = 1}^\infty \frac{(-1)^{n + 1}}{n} \;=\; 1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots
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| </math>
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| 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:
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| :<math>1 \,-\, \frac{1}{2} \,+\, \frac{1}{3} \,-\, \frac{1}{4} \,+\, \frac{1}{5} \,-\, \cdots \;=\; \ln 2.</math>
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| This formula is a special case of the [[Mercator series]], the [[Taylor series]] for the natural logarithm.
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| A related series can be derived from the Taylor series for the [[arctangent]]:
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| : <math>
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| \sum_{n = 0}^\infty \frac{(-1)^{n}}{2n+1} \;\;=\;\; 1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \cdots \;\;=\;\; \frac{\pi}{4}.
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| </math>
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| This is known as the [[Leibniz formula for pi]].
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| ===General harmonic series===
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| The '''general harmonic series''' is of the form
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| :<math>\sum_{n=0}^{\infty}\frac{1}{an+b} ,\!</math>
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| where ''<math>a \ne 0</math>'' and ''<math>b</math>'' are real numbers.
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| By the [[comparison test]], all general harmonic series diverge.
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| <ref>
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| Art of Problem Solving:
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| [http://www.artofproblemsolving.com/Wiki/index.php/Harmonic_series#General_Harmonic_Series "General Harmonic Series"] | |
| </ref>
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| ===''P''-series===
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| A generalization of the harmonic series is the '''''p''-series''' (or '''hyperharmonic series'''), defined as:
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| :<math>\sum_{n=1}^{\infty}\frac{1}{n^p},\!</math>
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| 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]] evaluated at ''p''.
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| ===φ-series===
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| For any convex, real-valued function φ such that
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| :<math>\limsup_{u\to 0^{+}}\frac{\varphi(\frac{u}{2})}{\varphi(u)}< \frac{1}{2} </math>
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| the series {{nowrap|1=∑<sub>''n''≥1</sub> φ(''n''<sup>−1</sup>)}} is convergent.
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| ===Random harmonic series===
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| The random harmonic series
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| :<math>\sum_{n=1}^{\infty}\frac{s_{n}}{n},\!</math>
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| where the ''s''<sub>''n''</sub> are [[statistical independence|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 [[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>"Random Harmonic Series", ''American Mathematical Monthly'' 110, 407-416, May 2003</ref><ref>[http://www.stat.ualberta.ca/people/schmu/preprints/rhs.pdf Schmuland's preprint of ''Random Harmonic Series'']</ref> 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 {{gaps|0.124|999|999|999|999|999|999|999|999|999|999|999|999|999|764…}}, differing from 1/8 by less than 10<sup>−42</sup>. 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 <math>C_2</math><ref>Weisstein, Eric W. “Infinite Cosine Product Integral.” From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html accessed 11/14/2010</ref> divided by π.
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| ===Depleted harmonic series===
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| {{main|Kempner series}}
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| 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>[http://www.qbyte.org/puzzles/p072s.html Nick's Mathematical Puzzles: Solution 72<!-- Bot generated title -->]</ref> In fact when terms containing any particular string of digits are removed the series converges.
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| ===Relation with Gamma Function===
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| The sum of harmonic progression is mathematically related to derivative of gamma function
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| :<math> 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots\, \frac{1}{n} </math> <math> =</math> <math> \left(\frac{{\rm d}}{{\rm d}x}\,\ln(\Gamma(x+1))\right)_{x=n} +\, \gamma.</math>
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| Here 'x' is the number of terms up to which sum is taken. <math> \gamma </math> is Euler–Mascheroni constant.
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| ===Sum of Harmonic series as an infinite series===
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| Harmonic series can be expressed as an infinite series as shown. This is known as Praliya's series. It was given by Neetesh Praliya.
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| :<math>
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| \begin{align}
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| 1 \,+\, \frac{1}{2} \,+\, \frac{1}{3} \,+\, \frac{1}{4} \,+\, \frac{1}{5} \,+\, \cdots\, + \,\frac{1}{n} &= 1 + \left( \sum_{m=1}^\infty \frac{1}{n^{m+1}} \left( \sum_{r=1}^{n-1}r^m \right) \right)\\
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| &= 1 + \frac{1}{n^2}\sum_{r=1}^{n-1}r + \frac{1}{n^3}\left( \sum_{r=1}^{n-1}r^2 \right) + \frac{1}{n^4}\left( \sum_{r=1}^{n-1}r^3 \right) + \frac{1}{n^5}\left( \sum_{r=1}^{n-1}r^4 \right) + \cdots\\
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| &= 1 + \frac{1 + 2 + \cdots + n-1}{n^2} + \frac{1^2 + 2^2 + \cdots + (n-1)^2}{n^3} + \frac{1^3 + 2^3 + \cdots + (n-1)^3}{n^4} + \cdots
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| \end{align}
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| </math>
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| ==See also==
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| {{Commons category|Harmonic series}}
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| * [[Complex logarithm]]
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| * [[Harmonic progression (mathematics)|Harmonic progression]]
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| * [[Riemann hypothesis#Growth rates of multiplicative functions|Lagarias's theorem]]
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| ==References==
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| {{reflist|30em}}
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| ==External links==
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| * {{springer|title=Harmonic series|id=p/h046540}}
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| <!-- * [http://eom.springer.de/H/h046540.htm "Harmonic Series" at Springer Encyclopaedia of Mathematics] -->
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| * [http://prairiestate.edu/skifowit/harmapa.pdf "The Harmonic Series Diverges Again and Again"], ''The AMATYC Review'', 27 (2006), pp. 31–43. Many proofs of divergence of harmonic series.
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| * {{MathWorld|title=Harmonic Series|urlname=HarmonicSeries}}
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| [[Category:Divergent series]]
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