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# Homework Assignment 22

Assigned Tuesday March 22; due Friday April 1, 2PM, at SS 1071

this document in PDF: HW.pdf

Required reading. All of Spivak's Chapters 22 and 23.

To be handed in. From Spivak Chapter 23: Problems 1 (parts divisible by 4), 12, 23.

Recommended for extra practice. From Spivak Chapter 23: Problems 1 (the rest), 5, 20, 21.

In class review problem(s) (to be solved in class on Thursday March 31):

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

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

Just for fun. 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).

Date: Sun, 20 Mar 2005 21:53:48 -0500

Dr. Bar-Natan: