Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 11 - Infinite Sequences and Series - 11.1 Sequences - 11.1 Exercises: 2


(a) A convergent sequence is a sequence that has a finite limit (i.e. $ \lim_{n \to \infty} a_n=L$). Examples: $$ a_n = \frac{1}{n},\space\space b_n = \frac{n}{n+1} $$ (b) A divergent sequence is any sequence that doesn't converge. It can oscillate between two values or shoot off to $ \pm\infty $. Examples: $$ a_n=n,\space\space b_n=\sin(\pi n) $$

Work Step by Step

(a) $$\begin{aligned}\lim\limits_{n \to \infty}a_n = \lim\limits_{n \to \infty}\frac{1}{n} &= \frac{1}{\infty}\\&=0\end{aligned} $$ Which is a finite number, so $ a_n $ is a convergent sequence. $$ \begin{aligned}\lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty}\frac{n}{n+1}&=\lim\limits_{n \to \infty}\frac{1+1/n}{1}\\&=\frac{1+1/\infty}{1}\\&=\frac{1+0}{1}\\&=1 \end{aligned}$$ Which is a finite number, so $b_n$ is a convergent sequence. (b) $$ \begin{aligned}\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}n&=\infty\end{aligned}$$ Since $a_n$ doesn't have a finite limit (it shoots off to $+\infty$), it is a divergent sequence. $$ \lim\limits_{n \to \infty}b_n=\lim\limits_{n \to \infty}\sin(\pi n)$$ Since $\sin(\pi n)$ oscillates between $-1$ and $1$, it doesn't tend towards a single finite number, so $b_n$ is a divergent sequence.
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.