Harmonic series

In mathematics, the harmonic series is the series that results from the summation of the terms of the harmonic series . Their partial sums are also called harmonic numbers . These are used, for example, in questions of combinatorics and are closely related to the Euler-Mascheroni constant . Although the harmonic series is a null series, the harmonic series is divergent . ${\displaystyle 1,{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},{\tfrac {1}{5}},\dotsc }$ ${\displaystyle \gamma }$

definition

The -th partial sum of the harmonic series is called the -th harmonic number: ${\displaystyle n}$ ${\displaystyle H_{n}}$ ${\displaystyle n}$

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

The harmonic series is a special case of the general harmonic series with the summands , where here , see below. ${\displaystyle 1/k^{\alpha }}$ ${\displaystyle \alpha =1}$

The name harmonic series was chosen because each link is the harmonic mean of the two neighboring links: ${\displaystyle a_{k}={\tfrac {1}{k}}}$

${\displaystyle M_{\text{H}}(a_{k-1},a_{k+1})={\frac {2}{{\frac {1}{a_{k-1}}}+{\frac {1}{a_{k+1}}}}}={\frac {2}{k-1+k+1}}={\frac {1}{k}}=a_{k}}$

properties

Values ​​of the first partial sums

${\displaystyle {\begin{matrix}H_{1}&=&1\\\\H_{2}&=&{\frac {3}{2}}&=&1{,}5\\\\H_{3}&=&{\frac {11}{6}}&=&1{,}8{\overline {3}}\\\\H_{4}&=&{\frac {25}{12}}&=&2{,}08{\overline {3}}\\\\H_{5}&=&{\frac {137}{60}}&=&2{,}28{\overline {3}}\\\\\end{matrix}}\qquad \qquad {\begin{matrix}H_{6}&=&{\frac {49}{20}}&=&2{,}45\\\\H_{7}&=&{\frac {363}{140}}&=&2{,}59{\overline {285714}}\\\\H_{8}&=&{\frac {761}{280}}&=&2{,}717{\overline {857142}}\\\\H_{9}&=&{\frac {7129}{2520}}&=&2{,}828{\overline {968253}}\\\\H_{10}&=&{\frac {7381}{2520}}&=&2{,}928{\overline {968253}}\\\\\end{matrix}}}$

The denominator of is divisible by every prime power , i.e. also by with and for, according to Bertrand's postulate, by at least one odd prime number . In particular, is for no whole number (Theisinger 1915). More generally, it is true that there is no difference for an integer ( Kürschák 1918), which is again a special case of a theorem by Nagell 1923. ${\displaystyle H_{n}}$ ${\displaystyle p^{k}}$ ${\displaystyle n/2<p^{k}\leq n}$ ${\displaystyle 2^{k}}$ ${\displaystyle 2^{k}\leq n}$ ${\displaystyle n>2}$ ${\displaystyle H_{n}}$ ${\displaystyle n>1}$ ${\displaystyle H_{m}-H_{n}}$ ${\displaystyle m\neq n}$

Is a prime number, the counter is of after the set of Wolstenholme by divisible, is a Wolstenholme-prime, then even through . ${\displaystyle p\geq 5}$ ${\displaystyle H_{p-1}}$ ${\displaystyle p^{2}}$ ${\displaystyle p}$ ${\displaystyle p^{3}}$

Nicholas of Oresme

divergence

The harmonic series diverges towards infinity, as Nicholas of Oresme first demonstrated. You can see this by comparing it with a series that is less than or equal in each term ( minorant criterion ):

${\displaystyle {\begin{matrix}H_{n}&=\ 1\ +\ 1/2&+&(1/3+1/4)&+&(1/5+1/6+1/7+1/8)&+\ \cdots \ +\ 1/n\\&\geq \ 1\ +\ 1/2&+&(1/4+1/4)&+&(1/8+1/8+1/8+1/8)&+\ \cdots \ +\ 1/n\\&=\ 1\ +\ 1/2&+&1/2&+&1/2&+\ \cdots \ +\ 1/n\end{matrix}}}$

The sum of the last row exceeds any value if is large enough. The estimate is obtained more precisely ${\displaystyle n}$

${\displaystyle H_{2^{\ell }}\geq 1+\ell /2}$   For   ${\displaystyle \ell =0,1,2,\dots }$

Asymptotic development

The asymptotic development applies :

${\displaystyle {\begin{aligned}H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}&=\ln n+\gamma +{\frac {1}{2n}}-{\frac {1}{12n^{2}}}+{\frac {1}{120n^{4}}}-{\frac {1}{252n^{6}}}+{\frac {1}{240n^{8}}}-{\frac {1}{132n^{10}}}+{\mathcal {O}}\!\left({\frac {1}{n^{12}}}\right)\\&=\ln n+\gamma +{\mathcal {O}}\!\left({\frac {1}{n}}\right),\quad n\to \infty \end{aligned}}}$

Here denotes the natural logarithm , and the Landau symbol describes the behavior of the residual term of the expansion for . The mathematical constant ( gamma ) is called the Euler-Mascheroni constant and its numerical value is 0.5772156649 ... ${\displaystyle \ln n}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle n\to \infty }$ ${\displaystyle \gamma }$

Partial sums of the harmonic series with approximation ln  n  +  γ and estimate ln  n  + 1

Furthermore , if . ${\displaystyle H_{n}<\ln n+1}$ ${\displaystyle n\geq 2}$

Comparison of some partial sums with values ​​of the approximate formula H _n  ≈ ln  n  +  γ

n	H n (rounded)	Approximation (rounded)	Accuracy (rounded)
5	2.28	2.19	95.77%
10	2.93	2.88	98.32%
20th	3.60	3.57	99.31%
50	4.50	4.49	99.78%
100	5.19	5.18	99.90%
500	6.79	6.79	1 - 1 · 10 −4
1000	7.49	7.48	1 - 7 · 10 −5
10,000	9.79	9.79	1 - 5 · 10 −6

Integral representation

It applies

${\displaystyle \int _{0}^{1}{\frac {1-x^{n}}{1-x}}\,\mathrm {d} x=\int _{0}^{1}(1+x+\cdots +x^{n-1})\ \mathrm {d} x=1+{\frac {1}{2}}+\cdots +{\frac {1}{n}}=H_{n}}$.

This representation generalizes the -th harmonic number to complex values ​​for with . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \operatorname {Re} (n)>-1}$

Special values ​​of the generalized harmonic numbers are for example:

${\displaystyle H_{1/2}=2-2\ln 2}$

${\displaystyle H_{1/3}=3-{\frac {\pi }{2{\sqrt {3}}}}-{\frac {3}{2}}\ln 3}$

${\displaystyle H_{1/4}=4-{\frac {\pi }{2}}-3\ln 2}$

${\displaystyle H_{1/6}=6-{\frac {\pi {\sqrt {3}}}{2}}-{\frac {3}{2}}\ln 3-2\ln 2.}$

Generating function

If you develop the function around the development point 0 in a Taylor series , you get the harmonic numbers as coefficients: ${\displaystyle {\tfrac {1}{1-x}}\ln {\tfrac {1}{1-x}}}$

${\displaystyle {\frac {1}{1-x}}\ln {\frac {1}{1-x}}=\sum _{n=1}^{\infty }H_{n}x^{n},\quad |x|<1}$

This can easily be seen by taking the Cauchy product of the for absolutely convergent series of ${\displaystyle |x|<1}$

${\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\dotsb }$

and

${\displaystyle \ln {\frac {1}{1-x}}=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+\dotsb }$

forms.

Relationship to the digamma function

The -th harmonic number can be expressed by the digamma function and continued to complex values ​​for (if it is not a negative integer): ${\displaystyle n}$ ${\displaystyle \psi }$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle H_{n}=\psi (n+1)-\psi (1)={\frac {\Gamma '(n)}{\Gamma (n)}}+{\frac {1}{n}}+\gamma }$.

The gamma function , its derivative and the Euler-Mascheroni constant denote . ${\displaystyle \Gamma }$ ${\displaystyle \Gamma '}$ ${\displaystyle \gamma }$

Series of harmonic numbers

The following applies to the harmonic numbers:

${\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}=2\,\zeta (3)}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{n^{3}}}={\frac {\pi ^{4}}{72}}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{n^{4}}}=3\,\zeta (5)-{\frac {\pi ^{2}}{6}}\,\zeta (3)}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{n^{5}}}={\frac {\pi ^{6}}{540}}-{\frac {1}{2}}\,\zeta (3)^{2}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {H_{n}}{n^{6}}}=4\,\zeta (7)-{\frac {\pi ^{2}}{6}}\,\zeta (5)-{\frac {\pi ^{4}}{90}}\,\zeta (3)}$

Here denotes the Riemann zeta function . ${\displaystyle \zeta (s)}$

Application example

Unsupported boom above, schematic drawing below.

Blocks of the same type should be stacked in such a way that the top block protrudes as far as possible over the bottom one.

The picture shows an application of the harmonic series. If the horizontal distances between the blocks - proceeding from top to bottom - are chosen according to the harmonious series, the stack is just about stable. In this way, the distance between the top and bottom blocks gets the greatest possible value. The blocks have a length . The top brick is with its focus on the second brick at the position . The common center of gravity of stone-1 and stone-2 lies with , that of stone-1, stone-2 and stone-3 with , that of the -th stone with . The total length of the boom is thus: ${\displaystyle l_{0}}$ ${\displaystyle 1/2\cdot l_{0}=1/2\cdot 1\cdot l_{0}}$ ${\displaystyle 1/2\cdot 1/2\cdot l_{0}}$ ${\displaystyle 1/2\cdot 1/3\cdot l_{0}}$ ${\displaystyle n}$ ${\displaystyle 1/2\cdot 1/n\cdot l_{0}}$ ${\displaystyle L}$

${\displaystyle L={\frac {l_{0}}{2}}\sum _{k=1}^{n}{\frac {1}{k}}}$.

Each additional stone corresponds to a further summand in the harmonic series. Since the harmonic series can take on any size, if you continue it far enough, there is no fundamental limit to how far the top stone can overhang. The number of stones required increases very quickly with the desired overhang. For an overhang 2.5 times the length of the stone, around 100 stones will be required. In a real structure, this would place high demands on the dimensional accuracy of the stones.

Further examples of the application of the harmonic series are the collector problem and the problem of 100 prisoners .

Related ranks

The partial sums of the alternating harmonic series

The alternating harmonic series converges:

${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}\pm \cdots =\ln 2=0{,}69314{\text{ }}71805{\text{ }}59945{\text{ }}30941\ldots }$

The convergence follows from the Leibniz criterion , the limit value can be calculated with the Taylor expansion of the natural logarithm and the Abelian limit value theorem . It is namely and if you bet, you get the alternating harmonic series in the series expansion. ${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {x^{k}}{k}}}$ ${\displaystyle x=1}$

A general harmonic series is called

${\displaystyle S=\sum _{k=1}^{\infty }{\frac {1}{k^{\alpha }}}\ ,}$

it diverges for and converges for (see Cauchy's compression criterion ). Their n -th partial sums are also referred to as or . ${\displaystyle \alpha \leq 1}$ ${\displaystyle \alpha >1}$ ${\displaystyle H_{n}^{(\alpha )}}$ ${\displaystyle H^{(\alpha )}(n)}$

Example for (see Basel problem ): ${\displaystyle \alpha =2}$

${\displaystyle S=\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}=1{,}64493{\text{ }}40668{\text{ }}48226{\text{ }}43647\ldots }$

Example for : ${\displaystyle \alpha =4}$

${\displaystyle S=\sum _{k=1}^{\infty }{\frac {1}{k^{4}}}=1+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+\cdots ={\frac {\pi ^{4}}{90}}=1{,}08232{\text{ }}32337{\text{ }}11138{\text{ }}19151{\text{ }}\ldots }$

Example for : ${\displaystyle \alpha =2n}$

${\displaystyle S=\sum _{k=1}^{\infty }{\frac {1}{k^{2n}}}=1+{\frac {1}{2^{2n}}}+{\frac {1}{3^{2n}}}+{\frac {1}{4^{2n}}}+\cdots =(-1)^{n-1}\,{\frac {(2\pi )^{2n}}{2(2n)!}}\,\mathrm {B} _{2n}=\zeta (2n)}$

where the -th denotes Bernoulli number . ${\displaystyle \mathrm {B} _{2n}}$ ${\displaystyle 2n}$

If one also allows for complex numbers, one arrives at the Riemann zeta function . ${\displaystyle \alpha }$

Subharmonic series

Subharmonic series are created by omitting certain summands in the series formation of the harmonic series, for example only adding the reciprocal values ​​of all prime numbers :

${\displaystyle S=\sum _{k{\text{ prim}}}^{\infty }{\frac {1}{k}}}$

This sum also diverges ( Euler's theorem ).

A convergent series arises if one adds up only over the prime twins (or even prime triplets or prime quadruplets etc.); however, it is not known whether these are infinite series. The limit values ​​are called Brun's constants .

Further subharmonic series are the likewise convergent Kempner series .

literature

• Harro Heuser : Textbook of Analysis Part 1 . 5th edition. Teubner-Verlag, 1988, ISBN 3-519-42221-2 .

Web links

Wikibooks: Math for Non-Freaks: Harmonious Series  - Learning and teaching materials

• Eric W. Weisstein : Harmonic Series . In: MathWorld (English).
• Eric W. Weisstein : Harmonic Number . In: MathWorld (English).

Individual evidence

1. ^ Leopold Theisinger: Comment on the harmonic series , monthly books for mathematics and physics 26, 1915, pp. 132-134
2. ↑ József Kürschák : A harmonikus sorról (About the harmonic series), Mathematikai és physikai lapok 27, 1918, pp. 299-300 (Hungarian)
3. ^ Trygve Nagell : A property of certain sums , Videnskapsselskapet scrifter. I. Matematisk-Naturvidenskabelig Klasse 13, 1923, pp. 10-15
4. ↑ Borwein, D. and Borwein, JM "On an Intriguing Integral and Some Series Related to zeta (4)." Proc. Amer. Math. Soc. 123, 1191-1198, 1995.