Alternating series test

The alternating series test, also known as Leibniz's test, is a method used to determine the convergence of a series where the terms alternate in sign. Specifically, it applies to a series where each term is given by an = (−1)n+1|an| for all n ∈ N. If this condition is met, the alternating series test can be used to determine whether the series converges or diverges. The test is named after Gottfried Wilhelm Leibniz, who first described it in a letter written in 1705.From: Sum of alternating-like series as definite integrals [2022], A Course in Ordinary Differential Equations [2019], Advanced Calculus [2019]

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Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018

Dan Zwillinger

If |anan+1|=1+pn+Annq $ |\frac{{a_{n} }}{{a_{n + 1} }}| = 1 + \frac{p}{n} + \frac{{A_{n} }}{{n^{q} }} $ , for sufficiently large n, where q > 1 and the sequence {An} is bounded, mpote nte series is absolutely convergent if and only ua p > 1.Alternating series test

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Published in John Srdjan Petrovic, Advanced Calculus, 2020

John Srdjan Petrovic

so the radius of convergence is R = 1. Further, when x = 1 we have ∑n=0∞(−1)n3n+1 which converges by Alternating Series Test. Thus, the desired result is f(1).

Sum of alternating-like series as definite integrals

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Published in International Journal of Mathematical Education in Science and Technology, 2022

Sergio A. Carrillo

The goal of the present note is to contribute to this point by summing the family of series where the sequence is k-periodic, i.e. , for all integers and satisfies We usually use two well-known methods. The first method, to establish the convergence of (1) is Dirichlet's test (Rudin, 1976, p. 70): if is a decreasing sequence of real numbers such that as , and a sequence of complex numbers with bounded partial sums, then converges. The case corresponds to Leibniz's test (also called alternating series test). Our second tool is Abel's continuity theorem for power series (Rudin, 1976, p. 174): if the series of complex numbers converges, then the power series converges for and We will prove that this limit reduces the sum of (1) to the definite integral which is finite since the factor 1−t can be eliminated from the numerator and denominator of fraction, thanks to the condition (2).

Alternating series test

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Numbers and Elementary Mathematics

Solutions and Answers to Selected Exercises

Sum of alternating-like series as definite integrals