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The harmonic series is
(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 is called the th harmonic of tone with wavelength 1�� .
Theorem. The harmonic series is divergent.
First proof: If is a partial sum of this series and an arbitrary real number, then for and we have
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��
Since
we get
(2) |
But , consequently
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���
For an exposition of��a proof that the harmonic series diverges given by Jakob Bernoulli in 1689 consult [1] .
The th partial sum of the harmonic series is called the th harmonic number . Consequently all lower bounds for harmonic number growing to the infinity imply the divergence of the harmonic series.
In 1914 A.J.Kempner��[2]��proved the following interesting result:
Theorem: If denotes the set of integers whose decimal representation has no 9's in its digits then is convergent and its sum is .
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
and is always divergent.
Euler showed that the sum over all primes diverges, and Dirichlet that the related sum over all primes of the form with coprime also diverges.
A famous problem of Erdõs asks:��Is it true that whenever ��is a sequence of positive integers��such that diverges, then the sequence contains arbitrarily long arithmetic progressions?
[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. Retrieved 2024/12/29 from Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, web-page http://www.cs.cas.cz/portal/AlgoMath/MathematicalAnalysis/InfiniteSeriesAndProducts/InfiniteSeries/SeriesWithNonNegativeTerms/HarmonicSeries.htm