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The complete sequences include: The sequence of the number 1 followed by the prime numbers (studied by S. S. Pillai [3] and others); this follows from Bertrand's postulate. [1] The sequence of practical numbers which has 1 as the first term and contains all other powers of 2 as a subset. [4] (sequence A005153 in the OEIS)
When every term of a series is a non-negative real number, for instance when the terms are the absolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound ...
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.
A number that has the same number of digits as the number of digits in its prime factorization, including exponents but excluding exponents equal to 1. A046758: Extravagant numbers: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, ... A number that has fewer digits than the number of digits in its prime factorization (including ...
More generally, any sequence of rational numbers that converges to an irrational number is Cauchy, but not convergent when interpreted as a sequence in the set of rational numbers. Metric spaces that satisfy the Cauchy characterization of convergence for sequences are called complete metric spaces and are particularly nice for analysis.
Beginning of the Fibonacci sequence on a building in Gothenburg. In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms.
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A one-dimensional optimization method, called the Fibonacci search technique, uses Fibonacci numbers. [75] The Fibonacci number series is used for optional lossy compression in the IFF 8SVX audio file format used on Amiga computers. The number series compands the original audio wave similar to logarithmic methods such as μ-law. [76] [77]