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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.
Then the triangle is in Euclidean space if the sum of the reciprocals of p, q, and r equals 1, spherical space if that sum is greater than 1, and hyperbolic space if the sum is less than 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.
The Erdős–Graham conjecture follows from this result by showing that one can find an interval of this form in which the sum of the reciprocals of all smooth numbers is at least ; therefore, if the integers are -colored there must be a monochromatic subset satisfying the conditions of Croot's theorem.
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: = = + + + + +.. The first terms of the series sum to approximately +, where is the natural logarithm and is the Euler–Mascheroni constant.
The set of all finite sums over D is often denoted as FS(D). Slightly more generally, for a sequence of natural numbers (n i), one can consider the set of finite sums FS((n i)), consisting of the sums of all finite length subsequences of (n i). A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset ...
Littlewood stated the principles in his 1944 Lectures on the Theory of Functions [1] as: . There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function (of class L p) is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.
In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
where A is a finite nonempty subset of a field F, and p(F) is a prime p if F is of characteristic p, and p(F) = ∞ if F is of characteristic 0. Various extensions of this result were given by Noga Alon , M. B. Nathanson and I. Ruzsa in 1996, [ 11 ] Q. H. Hou and Zhi-Wei Sun in 2002, [ 12 ] and G. Karolyi in 2004.