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In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
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.
By this construction, the function that defines the harmonic number for complex values is the unique function that simultaneously satisfies (1) H 0 = 0, (2) H x = H x−1 + 1/x for all complex numbers x except the non-positive integers, and (3) lim m→+∞ (H m+x − H m) = 0 for all complex values x.
Harmonic progression may refer to: Chord progression in music; Harmonic progression (mathematics) Sequence (music) This page was last edited on 28 ...
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.
In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians ; the solutions to which are given by eigenvalues corresponding to their modes of vibration.