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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, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors.For example, 21 is the product of 3 and 7 (the result of multiplication), and (+) is the product of and (+) (indicating that the two factors should be multiplied together).
The product-to-sum identities [28] or prosthaphaeresis formulae can be proven by expanding their right-hand sides using the angle addition theorems. Historically, the first four of these were known as Werner's formulas, after Johannes Werner who used them for astronomical calculations. [29]
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
The natural numbers 0 and 1 are trivial sum-product numbers for all , and all other sum-product numbers are nontrivial sum-product numbers. For example, the number 144 in base 10 is a sum-product number, because 1 + 4 + 4 = 9 {\displaystyle 1+4+4=9} , 1 × 4 × 4 = 16 {\displaystyle 1\times 4\times 4=16} , and 9 × 16 = 144 {\displaystyle 9 ...
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.
In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence ().As a consequence the partial sums of the series only consists of two terms of () after cancellation.
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