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Harmonic series

The harmonic series is

Underoverscript[∑, n = 1, arg3] 1/n .(1)

The name of the series originates in the fact that every term of the harmonic series is equal to the harmonic mean (a name introduced by Hippas of Metapont (ca. 450 BC)) of its preceding and following terms. Another explanation says that the name goes back to the fact that a tone of wavelength typeset structure is called the typeset structureth harmonic of tone with wavelength 1   .

Theorem. The harmonic series is divergent.

First proof: If typeset structure is a partial sum of this series and typeset structure an arbitrary real number, then for typeset structure and typeset structure we have

H _ n > (1 + 1/2) + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... + (1/(2^(m - 1) + 1) + ... + 1/2^m) > 1/2 + 2 1/4 + 4 1/8 + 8 1/16 + ... + 2^(m - 1) 1/2^m = m/2 > A,

that is, the sequence of partial sums diverges.

The above proof is due to  Nicole Oresme (c. 1323 - 1382)   and  presents a gem of medieval mathematics.

Second proof: Suppose that the harmonic series is convergent. Since it is a series with positive terms, it is absolutely convergent and arbitrary rearrangements of its terms does not influence its convergence behavior . This observation allows us to split it into two terms, one containing its even terms and that of odd terms  

Underoverscript[∑, n = 1, arg3] 1/n = Underoverscript[∑, n = 1, arg3] 1/(2 n) + Underoverscript[∑, n = 1, arg3] 1/(2 n - 1) .

Since

Underoverscript[∑, n = 1, arg3] 1/n = 2 Underoverscript[∑, n = 1, arg3] 1/(2 n),

we get

Underoverscript[∑, n = 1, arg3] 1/(2 n) = Underoverscript[∑, n = 1, arg3] 1/(2 n - 1) .(2)

But typeset structure, consequently

Underoverscript[∑, n = 1, arg3] 1/(2 n) < Underoverscript[∑, n = 1, arg3] 1/(2 n - 1)

which contradicts  (2), and therefore the assumption that  (1) is a convergent series is false.

Third proof: The divergency of the harmonic series can also be proved via integral test comparing it with the divergent improper integral   

∫ _ 1^∞ 1/x d x .

Fourth proof: Visit .

For an exposition of  a proof that the harmonic series diverges given by Jakob Bernoulli in 1689 consult [1] .

The typeset structureth partial sum typeset structure of the harmonic series is called the typeset structureth harmonic number . Consequently all lower bounds for harmonic number typeset structure growing to the infinity imply the divergence of the harmonic series.

In 1914 A.J.Kempner  [2]  proved the following interesting result:

Theorem: If typeset structure denotes the set of integers whose decimal representation has no 9's in its digits then typeset structure is convergent and its sum is typeset structure.

R. Honsberger   [3] , pp.  98--103,  gives a simple proof of this result and also some  references to related results for other missing digits.

The general harmonic series is defined by

Underoverscript[∑, n = 1, arg3] 1/(a n + b) ,       a, b ∈ R

and is always divergent.

Euler showed that the sum typeset structureover all primes diverges, and Dirichlet that the related sum typeset structure over all primes typeset structure of the form typeset structure with typeset structure coprime also diverges.

A famous problem of Erdõs asks:  Is it true that whenever typeset structure  is a sequence of positive integers  such that typeset structure diverges, then the sequence typeset structure contains arbitrarily long arithmetic progressions?

References

[1]  Dunham, W. (1987). The Bernoullis and the harmonic series. College Math. J., 18(1), 18-23.

[2]  Kempner, A. J. (1914). A Curious Convergent Series. American Mathematical Monthly, 21(1), 48-50.

[3]  Honsberger, R. (1976). Mathematical gems, II. Washington, DC: Math. Assoc. America.

Cite this web-page as:

Štefan Porubský: Harmonic series.

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