Limit comparison test

Infinite Series

Published in John Srdjan Petrovic, Advanced Calculus, 2020

Since the limit is a real number different from 0, the Limit Comparison Test guarantees that either both series converge or they both diverge. The series ∑n=1∞1nis the so-called Harmonic Series, known to be divergent. Therefore, ∑n=1∞n−12n2−3n+5 diverges as well.

Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables)

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Published in Kenneth B. Howell, Ordinary Differential Equations, 2019

Kenneth B. Howell

Under certain conditions, you can use the limit of the ratio of the consecutive terms of a single series to construct a geometric series that can serve as a second series in the above limit comparison test. That leads to a third test, which will be used near the end of Chapter 37.

Variable Selection for the Prediction of C[0,1]-Valued Autoregressive Processes using Reproducing Kernel Hilbert Spaces

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Journal Information

Published in Technometrics, 2019

Beatriz Bueno-Larraz, Johannes Klepsch

By Lemma 1 of Pumo (1998) we know that for some positive constants A1, A2, A3, where m is the sample size and 0 < r < 1 is given by assumption H1. By Borel–Cantelli, if the series in m whose terms are these probabilities is convergent for every ϵ > 0, we get the almost sure convergence stated in Equation (12). The sum is of order where Cϵ, Cr, Dϵ > 0 and these three series converge (for example, by the limit comparison test with ∑m− γ, γ > 1). Concerning (13), the same Lemma 1 of Pumo (1998) states that the bounds for these probabilities are equivalent but with m − 1 in place of m.

Limit comparison test

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Infinite Series

Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables)

Variable Selection for the Prediction of C[0,1]-Valued Autoregressive Processes using Reproducing Kernel Hilbert Spaces