A Short and Unified Proof of Kummer's Test

2018/02/27 by Sjödin, Tord
#00A30 #01A55 #40A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #History and Overview (math.HO)

paper · doi:10.48550/arxiv.1802.09858

Abstract

Kummer's test from 1835 states that the positive series ∑_n=1^∞ a_n is convergent if and only if there is a sequence \ B_n\₁^∞ of positive numbers such that B_n⋅ \fraca_n a_n+1 -B_n+1≥ 1 , for all sufficiently large n. We present an exact analysis and a short and unified proof of Kummer's test. The test has been applied to differential equations and studied in mathematical philosophy.

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