# Homework Assignment 22

Assigned Tuesday March 11; due Friday March 21, 2PM at SS 1071

this document in PDF: HW22.pdf

Required reading. All of Spivak Chapter 23. Also read (but don't do!) all exercises for that chapter -- just to get an impression for how intricate the various convergence tests and criteria can get.

To be handed in. From Spivak Chapter 23: 1 (parts divisible by 4), 12, 23 as well as the following question:

• Prove that the following sums diverge: (Hint: Use problem 20.)

• Prove that the following sums converge: (Hint: Use problem 20.)

Recommended for extra practice. From Spivak Chapter 23: 1 (the rest), 5, 20, 21 as well as the following question:

• In this question we always assume that and . Let's say that a sequence is much bigger'' than a sequence if . Likewise let's say that a sequence is much smaller'' than a sequence if . Prove that for every convergent series there is a much bigger sequence for which is also convergent, and that for every divergent series there is a much smaller sequence for which is also divergent. (Thus you can forever search in vain for that fine line between good and evil; it just isn't there).

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Dror Bar-Natan 2003-03-11