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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.
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.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.
With the exceptions of 1, 8 and 144 (F 1 = F 2, F 6 and F 12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). [56] As a result, 8 and 144 (F 6 and F 12) are the only Fibonacci numbers that are the product of other Fibonacci numbers. [57]
The first four partial sums of the series 1 + 2 ... used in mathematics to assign ... in the Euler–Maclaurin formula for the partial sums of a series.
Define r as follows: = | |, where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value). If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The last series is known as Mercator series, named after Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia). [14] Both of these series converge for | | <. (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.) [13]
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...