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The harmonic series is the infinite series = = + + + + + in which the terms are all of the positive unit fractions. It is a divergent series : as more terms of the series are included in partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit.
The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
This was proved by Leonhard Euler in 1737, [1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).
The series was first studied by A. J. Kempner in 1914. [3] The series is counterintuitive [1] because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 90. Baillie [4] showed that, rounded to 20 decimals, the actual sum is 22.92067 66192 64150 34816 (sequence A082838 in the OEIS).
This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula ψ ( n ) = H n − 1 − γ {\displaystyle \psi (n)=H_{n-1}-\gamma } where γ is the Euler–Mascheroni constant .
It is known, that the harmonic numbers are never integers except the case n=1. The same question can be posed with respect to the hyperharmonic numbers: are there integer hyperharmonic numbers? István Mező proved [5] that if r=2 or r=3, these numbers are never integers except the trivial case when n=1.
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
Harmonic series may refer to either of two related concepts: Harmonic series (mathematics) Harmonic series (music) This page was last edited on 28 ...
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