Chapter 8.2, Problem 75E

a.

To determine

To explain: Determine whether the statement “If

$\underset{n\to \infty }{\lim }{a}_n=1\text{ and }\underset{n\to \infty }{\lim }{b}_n=3,\text{ then }\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=3$”is true or not.

b.

To determine

To explain: Determine whether the statement “If

$\underset{n\to \infty }{\lim }{a}_n=0\text{ and }\underset{n\to \infty }{\lim }{b}_n=\infty ,\text{ then }\underset{n\to \infty }{\lim }{a}_n{b}_n=0$”is true or not.

c.

To determine

To explain: Determine whether the statement “The convergent sequences $\left\{a_n\right\}\text{ and }\left\{b_n\right\}$ differ in their first 100 terms, but $a_n=b_n\text{ for }n>100$. It follows that $\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{b}_n$” is true or not.

d.

To determine

To explain: Determine whether the statement “If $\left\{a_n\right\}\text{=}\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\cdots \right\}\text{ and }\left\{b_n\right\}=\left\{1,0,\frac{1}{2},0,\frac{1}{3},0,\cdots \right\}$, then $\underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{b}_n$” is true or not.

e.

To determine

To explain: Determine whether the statement “If $\left\{a_n\right\}$ converges, then. $\left\{{\left(-1\right)}^n{a}_n\right\}$ converges” is true or not.

f.

To determine

To explain: Determine whether the statement “If $\left\{a_n\right\}$ diverges, then $\left\{0.000001a_n\right\}$ diverges” is true or not.