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Homework Assignment 14

Assigned Tuesday January 18; due Friday January 28, 2PM, at SS 1071

this document in PDF: HW.pdf

Required reading. All of Spivak's chapters 14 and 15.

To be handed in. From Spivak Chapter 14: 11, 15, 21. From Chapter 15: 2 (odd parts).

Recommended for extra practice. From Spivak Chapter 14: 7, 19, 25, 28. From Chapter 15: 2 (even parts).

In class review problem(s) (to be solved in class this Thursday). Chapter 14 problem 25 parts (a) and (b): The limit $\lim_{N\to\infty}\int_a^Nf$, if it exists, is denoted by $\int_a^\infty f$ (or $\int_a^\infty f(x)dx$), and called an ``improper integral.''

(a) Determine $\int_1^\infty x^rdx$, if $r<-1$.

(b) Use Problem 13-15 to show that $\int_1^\infty 1/x\,dx$ does not exist.

Hint: What can you say about $\int_1^{2^n}1/x\,dx$?

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