Harmonic progression

It is emission infinite series with a harmonic progression (ちょうわきゅうすう British: harmonic series) in the mathematics

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

I say a saw. The wavelength of the harmonic overtone of the string which "harmony" (harmonics) of the name comes from the concept of the harmonic overtone in music and the thoroughbass, and vibrates is 1/2, 1/3, 1/4, of the basic wavelength of the string ... となっていることによる. Each clause of the harmonic progression becomes the harmonic average of the anteroposterior clause, and after all the term harmonic average comes from music again, too.

Table of contents

History

The first proof of a harmonic progression emitting it as a historical fact was a thing by Nicole オレーム of the 14th century, but there was an error in this. Right proof being accomplished later in the 17th century by Pietro mon gobiid, Johan Bernoulli, Jacob Bernoulli and others.

We had demand from the viewpoint of architecture in the harmonic progression historically. For the baroque in particular in a ground plan and the elevation was used to balance it or to establish the harmony relations of the structural details of interior decoration and the decoration of a church and the palace [¹].

Introduction

In the meaning that I display though the limit of the clause becomes 0 as for the harmonic progression, it is an intuitive series for the abecedarian. In other words, the infinite sum of progression to converge to 0 is shown not to have possibilities to necessarily converge to a limited price. A result against some paradox and intuition due to a harmonic progression emitting it is known.

For example, with the paradox called "green caterpillar ("worm on the rubber band") on the elastic" [²]. The contents "there is elastic (can grow endlessly), and a green caterpillar shall crawl over a string with 1 centimeter a minute of speed towards other edge, and is enlarged 1 meter to the elastic length uniformly (right after a green caterpillar crawled 1 centimeter exactly) every one minute by one end of the string, and, in other words, the green caterpillar only crawled on 1 centimeter from the initial point one minute later, but can the green caterpillar arrive at the edge of string of 1 meter when, actually, is at 4.5 centimeters of positions from the initial point, and repeat such a process though, actually, (because elastic was enlarged) will be at 2 centimeters of positions from the initial point, and only 1 centimeter crawls more from there two minutes later?." The answer "can arrive at it" against intuition. When assume the distance with the position where a green caterpillar is after starting point and T L_T centimeter; L_T

${\displaystyle L_{T}={\frac {T+1}{T}}{\bigl (}L_{T-1}+1{\bigr) },}$  (but${\displaystyle L_{0}=0}$ とする)

I am expressed in という 漸化式. When I solve this,

${\displaystyle L_{T}=(T+1)\sum _{n=1}^{T}{\frac {1}{n}}}$ 

となる. On the other hand, a green caterpillar being able to arrive at the endpoint because the length of the elastic after T is 100 (T +1) centimeter

${\displaystyle (T+1)\sum _{n=1}^{T}{\frac {1}{n}}\geq 100 (T+1)}$ 

Then, come;, in other words

${\displaystyle {\frac {1}{100}}\sum _{n=1}^{T}{\frac {1}{n}}\geq 1}$ 

となるときである. Because this series can raise it without limit if I make n big, the upper expression consists for enough big T. In other words, the green caterpillar may arrive at the edge. But it is necessary to extremely increase a price of n so that such a thing happens. Specifically, it is quite e¹⁰⁰ ≒ 10^43.429 in the sum of Σ of the left side of a go board taking the slightly from ln (T +1) big value to see it at later integral calculus judgment law... At last it may arrive at the edge in a minute.

Another example includes a thing it "is apparent to be able to pile it up on the border of the table, but can you load when a group of the identical domino was given it so that then table のへりを (how) projects?". If "there is domino enough, this intuitive result can let project as much as desired without limit" [³]; [⁴].

Diverging

I display the harmonic progression in regular infinite + ∞. As for the method to prove this fact, some well known things exist.

Comparison test

One of the methods indicating the divergence of the harmonic progression is to compare it with the different emission series. The value of the sum of harmonic progression is bigger than the second series whether each clause of the harmonic progression is bigger than the clause to support of the following second serieses because otherwise I agree.

${\displaystyle {\begin{aligned}&1+\left({\frac {1}{2}}\right)+\left({\frac {1}{3}}+{\frac {1}{4}}\right)+\left({\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}\right)+\left({\frac {1}{9}}+\right.\cdots \\[12pt]>{}&1+\left({\frac {1}{2}}\right)+\left({\frac {1}{4}}+{\frac {1}{4}}\right)+\left({\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}+{\frac {1}{8}}\right)+\left({\frac {1}{16}}+\right.\cdots \\[12pt]={}&1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{2}}+\cdots =\infty .\end{aligned}}}$ 

However, the friendship of the harmonic progression becomes infinite by a comparison test likewise because the value of the second series is infinite. I compare it in the proof mentioned above if I speak it more clearly

${\displaystyle \sum _{n=1}^{\,2^{k}\!}{\frac {1}{n}}>1+{\frac {k}{2}}}$ 

But, it is established for any regular integer k. This proof depends on Nicole オレーム, and is an extremity of the mathematics of the Middle Ages. This method is taught as a standard thing of the textbook-like proof now. The judgment method of Cauchy becomes the thing which generalized this method.

Integral calculus judgment method

 

I can show the emission of the harmonic progression by a comparison with a certain wide sense integral calculus. In this, I think about the rectangular meeting of the countable infinite unit with the area corresponding to each clause of the harmonic progression. The rectangle corresponding to the clause of the n joint shall have width 1, height 1/n. The total of these rectangular areas is a harmonic progression

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots}$ 

I agree in a の level. On the other hand, I think about curve y = 1/x, and the area under the part of x ∈ [1, ∞] integrates a wide sense

${\displaystyle \int _{1}^{\infty }{\frac {dx}{x}}=\infty}$ 

である. Because this area is completely covered by rectangles like the point, the total of the rectangular area becomes infinite likewise, too. If I say more,

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

But, it is expected that I was shown. Such a technique is called integral calculus judgment law generally.

Emission rate

The emission of the harmonic progression is very slow, and, for example, the friendship of 1,⁰43 first clauses is smaller than 100 [⁵]. This depends on partial harmony being logarithmic increase (English version). In particular

${\displaystyle \sum _{n=1}^{k}{\frac {1}{n}}=\ln k+\gamma +\varepsilon _{k}}$ 

But, it is managed. As for γ, ε_k nears 0 at a limit of k →∞ with the oiler マスケローニ fixed number here. This result depends on Euler.

Partial sum

The 第 n part sum of the harmonic progression to display

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

I am called the number of は 第 n harmony.

The differences between number of the harmony and ln n of the n joint converge to the fixed number of the oiler. The difference between the number of the harmony of the different number never becomes the integer. In addition, no harmony number is an integer except n = 1 [⁶].

Related series

Change harmonic progression

 

The partial harmony among first 14 of the change harmonic progression (for a guilt line). A state to near 2 natural logarithms (red line) is seen.

Series

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

I am known as は change harmonic progression (alternating harmonic series). I understand the convergence characteristics of this series from convergence judgment law of Leibniz. The sum of this series is in particular equal to 2 natural logarithms. In other words

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

But, it is managed. Of the メルカトール series that is Taylor series of the natural logarithm function as for this expression when is special.

Series to be related from the Taylor series of the reverse tangent function

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

But, I am led. This is known as a formula of π of Leibniz.

General harmonic progression

The general harmonic progression (general harmonic series) assumes a, b a real number; as a thing named a ≠ 0

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

It is a series expressed by という form. I can show that any general harmonic progression emits it by a comparison test [⁷].

p-series

Using real number p regular as for the thing called p - series (p-series) by the generalization of the harmonic progression

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

I am expressed by の form. It is a normal harmonic progression and displays the p-series at the age of p = 1. If use integral calculus judgment law and the judgment method of Cauchy; the p-series is p > I understand that I converge at the time of 1 by all means (the p-series of this time is called A harmonic progression (over-harmonic series)). On the contrary, I display it at the time of p≤1. p > At the time of 1, the value of the sum of p-series is equal to value ζ(p) in p of the Lehman zeta function.

φ-series

In actual number convex function φ

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

For a thing satisfying を, it is a series

∑_n≥1φ(n-¹)

は converges by all means.

Probability harmonic progression

Byron シュムランド (Byron Schmuland) of the Alberta University a probability harmonic progression (random harmonic series)

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

Want to study it about の property [⁸]; [⁹]. s_n of molecules is an independence distribution random variable line taking a value of +-1 with probability such as one of 1/2 each. I can show that friendship of this random variable converges almost surely using en:Kolmogorov three-series theorem [⁹]. シュムランド showed that the limit became the random variable with some interesting properties. The value in density function +-2 of the particularly random variable

0.124 999 999 999 999 999 999 999 999 999 999 999 999 999 764…

Well, this is smaller as 10^-42 than 1/8(=0.125). This probability is near 1/8, but the explanation of not agreeing is shown in the article of シュムランド. The close value of this probability

${\displaystyle {\frac {1}{\pi }}\int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos {\Bigl (}{\frac {x}{n}}{\Bigr) }dx}$    ^[10]

I am given で [⁹].

Deterioration harmonic progression

The details refer to "ケンプナー series"

The deterioration harmonic progression (depleted harmonic series) is a series to be provided in what the thing which 9 appears somewhere of the number of members of 十進表示 of the denominator removes all in the clauses of the harmonic progression. The deterioration harmonic progression converges; and the value is 22.92067... となる [¹¹]. In fact, the series to be provided in that way converges even if I removed any specific number line from 十進表示列.

Allied item

Wiki media Commons has the category in conjunction with the harmonic progression.

• Double bare logarithmic functions
• Theorem (English version) of the rag ria
• Harmonic progression

References

1. ^ George L. Hersey, Architecture and Geometry in the Age of the Baroque, p 11-12 and p37-51.
2. ^ Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, pp. 258–264, ISBN 978-0-201-55802-9.
3. ^ Sharp, R.T. (1954), "Problem 52 is Pi Mu Epsilon Journal: Overhanging dominoes" 411–412.
4. ^ Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, pp. 258–264, ISBN 978-0-201-55802-9.
5. ^ Sequence A082912 in the On-Line Encyclopedia of Integer Sequences
6. ^ http://mathworld.wolfram.com/HarmonicNumber.html
7. ^ Art of Problem Solving: "General Harmonic Series"
8. ^ "Random Harmonic Series", American Mathematical Monthly 110, 407-416, May 2003
9. ^ ^{a b c} Schmuland's preprint of Random Harmonic Series
10. ^ Weisstein, Eric W. "Infinite Cosine Product Integral." From MathWorld – a Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteCosineProductIntegral.html accessed 2014-11-09
11. ^ Nick's Mathematical Puzzles: Solution 72

• Weisstein, Eric W., "Harmonic Series" - MathWorld.
• "Harmonic Series" at Springer Encyclopaedia of Mathematics

Outside link

• "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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