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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.
[1] [3] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers. [4] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
Consider these situations as an example, the two-player zero-sum game pictured at right or above. The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C.
Z, +), the group of integers with addition is infinite; Non-discrete Lie groups are infinite. For example, (R, +), the group of real numbers with addition is an infinite group; The general linear group of order n > 0 over an infinite field is infinite
Infinite sets are so common, that when one considers finite sets, this is generally explicitly stated; for example finite geometry, finite field, etc. Fermat's Last Theorem is a theorem that was stated in terms of elementary arithmetic , which has been proved only more than 350 years later.
In mathematical analysis, Cesàro summation (also known as the Cesàro mean [1] [2] or Cesàro limit [3]) assigns values to some infinite sums that are not necessarily convergent in the usual sense. The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.
On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element. However, the definition of maximal and minimal elements is more general. In particular, a set can have many ...
The same definition can be used for series = whose terms are not numbers but rather elements of an arbitrary abelian topological group.In that case, instead of using the absolute value, the definition requires the group to have a norm, which is a positive real-valued function ‖ ‖: + on an abelian group (written additively, with identity element 0) such that: