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Tom Mike Apostol (/ ə ˈ p ɑː s əl / ə-POSS-əl; [1] August 20, 1923 – May 8, 2016) [2] was an American mathematician and professor at the California Institute of Technology specializing in analytic number theory, best known as the author of widely used mathematical textbooks.
Sequences is a mathematical monograph on integer sequences. It was written by Heini Halberstam and Klaus Roth, published in 1966 by the Clarendon Press, and republished in 1983 with minor corrections by Springer-Verlag. [1] [2] Although planned to be part of a two-volume set, [3] [4] the second volume was never published.
Illustration of the squeeze theorem When a sequence lies between two other converging sequences with the same limit, it also converges to this limit.. In calculus, the squeeze theorem (also known as the sandwich theorem, among other names [a]) is a theorem regarding the limit of a function that is bounded between two other functions.
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
"subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence.
[2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6] Additional proofs were published in the 17th century by Pietro Mengoli [2] [7] and by Jacob Bernoulli.
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 real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function.Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. [1]
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