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Cauchy condensation

Cauchy condensation

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this homework we'll learn about a new convergence test called Cauchy's Condensation Test. Suppose n-1 an is a series where the terms an are >0 and decreasing, so an+1 < an for all n. The Cauchy condensation test says that %=D1 Lan converges if and only if 2"a2n converges. n=1 n=1 This test generalizes the idea from Oresme's proof of the divergence of the harmonic series where you group terms together in groups of size 1, 2, 4, 8 etc. 1. By grouping the terms an together into groups of size 1, 2, 4, . ..., 2", argue that 2N+1 -1 E an < 2" azn. n=1 n=0 2. By taking the limit N → o, argue that if En-12" azn converges then so does E1 an. (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series E an.) n%=D1 %3D1 This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if E-1 2" azn diverges then so does E=1 an. You don't have to show this. %3D1
Transcribed Image Text:In this homework we'll learn about a new convergence test called Cauchy's Condensation Test. Suppose n-1 an is a series where the terms an are >0 and decreasing, so an+1 < an for all n. The Cauchy condensation test says that %=D1 Lan converges if and only if 2"a2n converges. n=1 n=1 This test generalizes the idea from Oresme's proof of the divergence of the harmonic series where you group terms together in groups of size 1, 2, 4, 8 etc. 1. By grouping the terms an together into groups of size 1, 2, 4, . ..., 2", argue that 2N+1 -1 E an < 2" azn. n=1 n=0 2. By taking the limit N → o, argue that if En-12" azn converges then so does E1 an. (Hint: apply the monotone convergence theorem to the sequence of partial sums of the series E an.) n%=D1 %3D1 This is half of the proof that the Cauchy condensation test works. A similar argument can be used to show that if E-1 2" azn diverges then so does E=1 an. You don't have to show this. %3D1
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