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

Answer

(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.
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