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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.
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .
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In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that = + + = = +,. an alternating series.. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), [1] and was later independently rediscovered by James Gregory in ...
Head First is a series of introductory instructional books to many topics, published by O'Reilly Media.It stresses an unorthodox, visually intensive, reader-involving combination of puzzles, jokes, nonstandard design and layout, and an engaging, conversational style to immerse the reader in a given topic.
Thinking in Java (ISBN 978-0131872486) is a book about the Java programming language, written by Bruce Eckel and first published in 1998. Prentice Hall published the 4th edition of the work in 2006. The book represents a print version of Eckel’s “Hands-on Java” seminar.
(in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms – the infinitude of both +1s and −1s allows any finite number of 1s or −1s to be prepended, by Hilbert's paradox of the Grand Hotel) is a permutation of Grandi's series in which each value in the rearranged series corresponds to a value that is at ...
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