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Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [14] There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests.
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.
Examples abound, one of the simplest being that for a double sequence a m,n: it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. [4] For example take a m,n = 2 m − n. in which taking the limit first with respect to n gives 0, and with respect to m gives ∞.
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x.
The proofs of the other cases with ... taking the limit with respect to ... It appears as Problem 70 in Pólya and Szegő (1925).
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [ 1 ] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences . [ 2 ]
It states that for a converging sequence the sequence of the arithmetic means of its first members converges against the same limit as the original sequence, that is () with implies (+ +) / . [ 1 ] [ 2 ] The theorem was found by Cauchy in 1821, [ 1 ] subsequently a number of related and generalized results were published, in particular by Otto ...