Chapter 9.1, Problem 46E

To determine

To Match:A given sequence to the corresponding graph or table.

Answer to Problem 46E

The sequence corresponds to the graph:

  

Explanation of Solution

Given Information: The sequence is defined as $a_n=\frac{3n+1}{n}$

Concept Used:If $\underset{x\to c}{\lim }f\left(x\right)=M$ and $\underset{x\to c}{\lim }g\left(x\right)=N$ , then

Sum Rule : $\underset{x\to c}{\lim }f\left(x\right)+g\left(x\right)=M+N$

Calculation:

Given Information: The sequence is defined as $a_n={(-1)}^n\frac{3n+1}{n+3}$

Concept Used: If $\underset{x\to c}{\lim }f\left(x\right)=M$ and $\underset{x\to c}{\lim }g\left(x\right)=N$ , then

• Sum Rule : $\underset{x\to c}{\lim }f\left(x\right)+g\left(x\right)=M+N$
• Quotient Rule : $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{M}{N}$ , if $\underset{x\to c}{\lim }\text{g}\left(x\right)\ne 0$
• Product Rule : $\underset{x\to c}{\lim }f\left(x\right)\cdot g\left(x\right)=M\cdot N$

Calculation: Divide the numerator and denominator of the expression by $n$ to write

  $\underset{n\to \infty }{\lim }\left[{(-1)}^n\frac{3n+1}{n+3}\right]=\underset{n\to \infty }{\lim }\left[{(-1)}^n\frac{\frac{3n}{n}+\frac{1}{n}}{\frac{n}{n}+\frac{3}{n}}\right]$

  $=\underset{n\to \infty }{\lim }\left[{(-1)}^n\frac{3+\frac{1}{n}}{1+\frac{3}{n}.}\right]$

  $=\left[\underset{n\to \infty }{\lim }{(-1)}^n\right]\cdot \left[\underset{n\to \infty }{\lim }\frac{\left(3+\frac{1}{n}\right)}{\left(1+\frac{3}{n}\right)}\right]$ (Product Rule)

  $=\left[\underset{n\to \infty }{\lim }{(-1)}^n\right]\cdot \left[\frac{\underset{n\to \infty }{\lim }3+\underset{n\to \infty }{\lim }\frac{1}{n}}{\underset{n\to \infty }{\lim }1+\underset{n\to \infty }{\lim }\frac{3}{n}}\right]$ (Sum Rule and Quotient Rule)

  $=\left[\underset{n\to \infty }{\lim }{(-1)}^n\right]\cdot \left(\frac{3+0}{1+0}\right)=3\times \underset{n\to \infty }{\lim }{(-1)}^n$

  $=\left(\begin{array}{l}+3\text{ if n}\to \infty \text{ is even}\\ -3\text{ if n}\to \infty \text{ is odd}.\end{array}\right)$

Therefore, the sequence does not converge.

The appropriate graph, from the given choices, in which $a_n$ diverges (oscillates between $+3$ and $-3$ ) as $n$ becomes large is