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In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick ...
The FBISE was established under the FBISE Act 1975. [2] It is an autonomous body of working under the Ministry of Federal Education and Professional Training. [3] The official website of FBISE was launched on June 7, 2001, and was inaugurated by Mrs. Zobaida Jalal, the Minister for Education [4] The first-ever online result of FBISE was announced on 18 August 2001. [5]
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Loosely, with this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution. More precisely, the distribution of the associated random variable in the sequence becomes arbitrarily close to a specified fixed distribution.
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
The archetypical example of a filter is the neighborhood filter at a point in a topological space (,), which is the family of sets consisting of all neighborhoods of . By definition, a neighborhood of some given point is any subset whose topological interior contains this point; that is, such that . Importantly, neighborhoods are not required to be open sets; those are called open ...