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Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (a n) (with n running from 1 to infinity understood) the distance between a n and x approaches 0 as n → ∞, denoted
Download as PDF; Printable version; ... note that a 1 − a 2 is a lower bound of the monotonically decreasing sequence S 2m+1, ... Calculus. Vol. 1 (2nd ed.). ...
The first volume deals with determinate equations, while the second part deals with Diophantine equations. The last section contains a proof of Fermat's Last Theorem for the case n = 3, making some valid assumptions regarding Q ( − 3 ) {\displaystyle \mathbb {Q} ({\sqrt {-3}})} that Euler did not prove.
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus.
[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 second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically , but this condition is not necessary for convergence.
Download as PDF; Printable version; In other projects ... is a sequence, then it converges if and only if ... [2] References ...
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