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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.
A series or, redundantly, an infinite series, is an infinite sum. It is often represented as [ 8 ] [ 15 ] [ 16 ] a 0 + a 1 + a 2 + ⋯ or a 1 + a 2 + a 3 + ⋯ , {\displaystyle a_{0}+a_{1}+a_{2}+\cdots \quad {\text{or}}\quad a_{1}+a_{2}+a_{3}+\cdots ,} where the terms a k {\displaystyle a_{k}} are the members of a sequence of numbers ...
The explication was written to remedy apparent weaknesses in the logarithmic series [6] [infinite series for (+)] , [7] that had become republished due to Nicolaus Mercator, [6] [8] or through the encouragement of Isaac Barrow in 1669, to ascertain the knowing of the prior authorship of a general method of infinite series.
Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product: (;) = = (). This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q.
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
The mathematical constant e can be represented in a variety of ways as a real number.Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction.
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .