Harmonic series (mathematics)

The harmonic series is in mathematics series, produced by summation of the members of the harmonic sequence. Your partial sums are also called harmonic numbers. These are used for example in topics of combinatorics and are closely related to the Euler - Mascheroni constant. Although the harmonic sequence is a null sequence, the harmonic series is divergent.

  • 4.1 subharmonic series


The - th partial sum of the harmonic series is called the - th harmonic number:

The harmonic series is a special case of the general harmonic series with the summand, in which case, see below.

The name of harmonic series was chosen because each element is the harmonic mean of the two adjacent links:


Values ​​of the first partial sums

The denominator of is by any prime power with divisible, so by with and for after Bertrand 's postulate by at least one odd prime. In particular, for any integer ( Theis Inger 1915). General terms, there is no difference for an integer ( Kürschák 1918), this is in turn a special case of a set of Nagell 1923.

Is a prime number, then the numerator of by the theorem of Wolstenholme divisible by, is a Wolstenholme prime, then even by.


The harmonic series diverges to infinity, as was first proved by Nicholas Oresme. This can be seen by comparison to a series in each member is less than or equal to (minor comparison test ):

The sum of the last line exceeds any value if it is sufficiently large. Specifically, we obtain the estimate

Asymptotic Expansion

It is the asymptotic expansion:

Herein, the natural logarithm, and the Landau - symbol indicates the behavior of the remaining terms of the development. The mathematical constant (gamma ) is called the Euler - Mascheroni constant and its numerical value is 0.5772156649 ...

Furthermore applies if.

Integral representation

It is

This representation generalizes the - th harmonic number with complex values ​​.

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

Generating function

If we expand the function around the expansion point 0 in a Taylor series, we obtain the harmonic numbers as coefficients:

This one is easy to see by the Cauchy product of absolutely convergent series one of



Relationship with the digamma function

The - th harmonic number can be expressed by the digamma function and to complex values ​​continue (if not negative integer):

It denotes the gamma function, its derivative and the Euler - Mascheroni constant.

Series on harmonic numbers

It applies to the harmonic numbers:

Here denotes the Riemann zeta function.

Example of use

Similar blocks to be stacked so that the top block extends as far as possible on the bottom.

The picture shows an application of the harmonic series. If the horizontal distances of the blocks - advancing from top to bottom - chosen according to the harmonic series, so the stack is just stable. In this way the distance between the uppermost and lowermost block receives the greatest possible value. The blocks have a length. The top block is located with its focus on the second stone at the position. The common center of gravity of stone 1 and stone -2 is, at the stone -1, stone and stone 2 - 3, which of th stone at. The total length of the boom is thus:.

Each additional stone corresponds to a further term in the harmonic series. Since the harmonic series can take arbitrarily large values ​​, if they continues far enough, there is no fundamental limit on how far the top piece can hang. The number of stones needed increases, however, very quickly with the desired overhang. On the table above, one can read off that about 100 stones are needed for an overhang in 2.5x brick length. In a real system, this already high demands would be placed on the dimensional accuracy of the stones.

Another example of the use of the harmonic series is the collector problem.

Related series

The alternating harmonic series converges:

The convergence follows from the Leibniz criterion, the limit can be calculated with the Taylor expansion of the natural logarithm.

As a general harmonic series is defined as

It diverges for and converges (see Cauchy compression criterion).




Where the- th Bernoulli number called.

Allowed for complex numbers, you get to the Riemann zeta function.

Subharmonic series

Subharmonic series are created by leaving out certain summands in the row formation of the harmonic series, such sums only the reciprocals of all prime numbers:

This sum diverges also. A convergent series arises when one sums only over the prime twins ( or even triplets prime or prime quadruplets, etc.); However, it is not known whether this is an infinite series. The limits are called Brun's constants.

Other sub-harmonic series are also convergent Kempner series.