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The Riemann zeta function is defined for real > by the convergent series = = = + + +, which for = would be the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers except x = 1 {\displaystyle x=1} , where the extended function has a simple pole .
In mathematics, the n -th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n -th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: Here, ⌊·⌋ represents the floor function.
Harmonic mean. In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is normally only used for positive arguments. [1] It is the most appropriate average for rates such as speeds. [2][3] The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers.
The digamma function satisfies the recurrence relation. Thus, it can be said to "telescope" 1/x, for one has. where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula. where γ is the Euler–Mascheroni constant.
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. 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.
Kempner series. The Kempner series[ 1 ][ 2 ]: 31–33 is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum. where the prime indicates that n takes only values whose decimal expansion has no nines.
Stirling's approximation. Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .