Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:

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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.)
  $$\sum_{n=1}^\infty \frac{1}{n}; \qquad \sum_{n=2}^\infty \frac{1}{n(\log n)}; \qquad \sum_{n=3}^\infty \frac{1}{n(\log n)(\log\log n)};$$
  $$\sum_{n=16}^\infty \frac{1}{n(\log n)(\log\log n)(\log\log\log n)}; \quad\ldots$$

• Prove that the following sums converge: (Hint: Use problem 20.)
  $$\sum_{n=1}^\infty \frac{1}{n^{1.01}}; \qquad \sum_{n=2}^\infty \f... ...g n)^{1.01}}; \qquad \sum_{n=3}^\infty \frac{1}{n(\log n)(\log\log n)^{1.01}};$$
  $$\sum_{n=16}^\infty \frac{1}{n(\log n)(\log\log n)(\log\log\log n)^{1.01}}; \quad\ldots$$

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 $a_n>0$ and $b_n>0$. Let's say that a sequence $a_n$ is ``much bigger'' than a sequence $b_n$ if $\lim_{n\to\infty}a_n/b_n=\infty$. Likewise let's say that a sequence $a_n$ is ``much smaller'' than a sequence $b_n$ if $\lim_{n\to\infty}a_n/b_n=0$. Prove that for every convergent series $\sum b_n$ there is a much bigger sequence $a_n$ for which $\sum a_n$ is also convergent, and that for every divergent series $\sum b_n$ there is a much smaller sequence $a_n$ for which $\sum a_n$ 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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