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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.
The infinite sequence of additions expressed by a series cannot be explicitly performed in sequence in a finite amount of time. However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series.
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.
With some exceptions, an expression can only have components that themselves have components of that type. For example, algebraic expressions can only have constants that are themselves algebraic, and finite sums and products in algebraic expressions can only have algebraic arguments. This template expands to a table.
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 sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum: For example, consider the sum: 2 + 5 + 8 + 11 + 14 = 40 {\displaystyle 2+5+8+11+14=40}
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
The set of all finite sums over D is often denoted as FS(D). Slightly more generally, for a sequence of natural numbers (n i), one can consider the set of finite sums FS((n i)), consisting of the sums of all finite length subsequences of (n i). A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset ...
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