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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.
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
It was the first calculator that could perform all four basic arithmetic operations. [3] Its intricate precision gearwork, however, was somewhat beyond the fabrication technology of the time; mechanical problems, in addition to a design flaw in the carry mechanism, prevented the machines from working reliably. [4] [5]
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.
300 BC — Mesopotamia, the Babylonians invent the earliest calculator, the abacus. [ 2 ] c. 300 BC — Indian mathematician Pingala writes the “Chhandah-shastra”, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a binary numeral system , along with the first use of Fibonacci ...
[1] [2] Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis.
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
Farey sequences are named after the British geologist John Farey, Sr., whose letter about these sequences was published in the Philosophical Magazine in 1816. [5] Farey conjectured, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours.