Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 10: Infinite Sequences and Series - Questions to Guide Your Review - Page 635: 1

Answer

See below.

Work Step by Step

$\text{ What is an infinite sequence? }$ An infinite sequence of numbers is a function whose domain is the set of positive integers. We denote the sequence $\{f(n)|n\in \mathbb{N}\}$ as $\{a_{n}\}$ where $f(n)=a_{n}.$ In simple terms, a list of numbers for which the first is $a_{1}$, second is $a_{2}$, nth is $a_{n}$ and there is a term corresponding to every natural number (which is why it is an infinite sequence). $\text{What does it mean for such a sequence to converge? To diverge?}$ The sequence $\{a_{n}\}$ converges to the number $L$ if for every $\epsilon\gt 0$ there exists an integer $N$ such that for all $n\gt N$ we have $ |a_{n}-L| \lt\epsilon.$ If no such number $L$ exists, then the sequence $\{a_{n}\}$ diverges. If $\{a_{n}\}$ converges to $L$, we write $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$ and call $L$ the $limit$ of the sequence. $\text{Give examples.}$ Examples of convergent sequences: $a_{n}=1\qquad \left\{1,1,1,...\right\}$ converges to 1 $a_{n}=\displaystyle \frac{1}{n}\qquad \left\{1,1/2,1/3,...\right\}$ converges to 0 Examples of divergent sequences $a_{n}=n\qquad \left\{1,2,3,...\right\} \quad$ $a_{n}=(-1)^{n}\qquad \left\{-1,1.-1,1,...\right\}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.