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In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: ... also work in any other normed vector space [6] ...
In general: + = + (())where both terms on the right are generalized pentagonal numbers and the first term is a pentagonal number proper (n ≥ 1).This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition.
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.
It concerns sequences of integers in which each term is obtained from the previous term as follows: if a term is even, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence.
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression.
The term Fibonacci sequence is also applied more generally to any function from the integers to a field for which (+) = + (+).These functions are precisely those of the form () = () + (), so the Fibonacci sequences form a vector space with the functions () and () as a basis.
The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. [ 3 ] [ 4 ] [ 5 ] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci , who introduced the sequence to Western ...
In the most common setting, the terms come from a commutative ring, so that the formal power series can be added term-by-term and multiplied via the Cauchy product. In this case the algebra of formal power series is the total algebra of the monoid of natural numbers over the underlying term ring. [76]