|
Convergence Proof Techniques
Convergence proof techniques are canonical components of mathematical proofs that limit of a sequence, sequences or limit of a function, functions converge to a finite limit (mathematics), limit when the argument tends to infinity. There are many types of series and Modes of convergence (annotated index), modes of convergence requiring different techniques. Below are some of the more common examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques. The links below give details of necessary conditions and generalizations to more abstract settings. The convergence of series is already covered in the article on convergence tests. Convergence in R''n'' It is common to want to prove convergence of a sequence f:\mathbb\rightarrow \mathbb^n or function f:\mathbb\rightarrow \mathbb^n, where \mathbb and \mathbb refer to the natural numbers and the real numbers, and convergence is with respect to the Euclidean norm, , , \cdot, , _2. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
