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Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
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.
[1] [2] Every term of the harmonic series after the first is the harmonic mean of the neighboring terms, so the terms form a harmonic progression; the phrases harmonic mean and harmonic progression likewise derive from music. [2] Beyond music, harmonic sequences have also had a certain popularity with architects.
The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.
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.
A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five of these are 1, 6, 28, 140, and 270. It is not known whether any harmonic divisor numbers (besides 1) are odd, but there are no odd ones less than 10 24. The sum of the reciprocals of the divisors of a perfect number is 2.
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is and the common difference of successive members is , then the -th term of the sequence is given by
In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence (). As a consequence the partial sums of the series only consists of two terms of ( a n ) {\displaystyle (a_{n})} after cancellation.