Leta(N) be such that (a(N)) is a decreasing sequence of strictly positive N=1 Numbers. IF S(N) denotes the Nth partial sum, show (by grouping the terms IN S(2) in two different ways) that: (a(1) + Za(2)+...+ 2a (2")) ≤ (a (1) + Za(z) + ... + 2Na (2N-1)) + a (2"). Use these inequalities to show that a(N) converges if and only if {2^a (2") converges. Use the Cauchy Condensation Test. N=1

Leta(N) be such that (a(N)) is a decreasing sequence of strictly positive N=1 Numbers. IF S(N) denotes the Nth partial sum, show (by grouping the terms IN S(2) in two different ways) that: (a(1) + Za(2)+...+ 2a (2")) ≤ (a (1) + Za(z) + ... + 2Na (2N-1)) + a (2"). Use these inequalities to show that a(N) converges if and only if {2^a (2") converges. Use the Cauchy Condensation Test. N=1

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 33E

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