True or False A sequence is a function whose domain is the set of positive real numbers.

True or False If the sequence $\{ s_{n}\}$ is convergent, then $\lim\limits_{n\rightarrow \infty}s_{n}=0.$

True or False If $f(x)$ is a related function of the sequence $\{ s_{n}\}$ and there is a real number $L$ for which $\lim\limits_{x\,\rightarrow \,\infty }\,f(x) =L$, then $\{ s_{n}\}$ converges.

Multiple Choice If there is a positive number $K$ for which $\vert s_{n}\vert \leq K$ for all integers $n\geq 1$, then $\{ s_{n}\}$ is [(a) increasing, (b) bounded, (c) decreasing, (d) convergent.]

True or False A bounded sequence is convergent.

True or False An unbounded sequence is divergent.

True or False A sequence $\{ s_{n}\}$ is decreasing if and only if $s_{n}\leq s_{n+1}$ for all integers $n\geq 1.$

True or False A sequence must be monotonic to be convergent.

True or False To use an algebraic ratio to show that the sequence $\{ s_{n}\}$ is increasing, show that $\dfrac{s_{n+1}}{s_{n}}\geq 0$ for all $n\geq 1$.

Multiple Choice If the derivative of a related function $f$ of a sequence $\{ s_{n}\}$ is negative, then the sequence $\{ s_{n}\}$ is [(a) bounded, (b) decreasing, (c) increasing, (d) convergent.]

True or False When determining whether a sequence $\{ s_{n}\}$ converges or diverges, the beginning terms of the sequence can be ignored.

True or False Sequences that are both bounded and monotonic diverge.

$s_{n}=\dfrac{n+1}{n}$

$s_{n}=\dfrac{2}{n^{2}}$

$s_{n}=\ln n$

$s_{n}=\dfrac{n}{\ln (n+1)}$

$s_{n}=\dfrac{(-1)^{n+1}}{2n+1}$

$s_{n}=\dfrac{1-(-1)^{n}}{2}$

$s_{n}=\left\{ \begin{array}{@{}l@{\quad}ll} (-1)^{n+1} & \hbox{if} & n~\hbox{is even} \\ 1 & \text{if} & n\hbox{ is odd} \end{array} \right.$

$s_{n}=\left\{ \begin{array}{@{}l@{\quad}ll} n^{2}+n & \text{if} & ~n~\hbox{is even} \\ 4n+1 & \text{if} & ~n~\hbox{is odd} \end{array} \right.$

$s_{n}=\dfrac{n!}{2^{n}}$

$s_{n}=\dfrac{n!}{n^{2}}$

$2, 4, 6, 8, 10, \ldots$

$1, 3, 5, 7, 9, \ldots$

$2,~4,~8,~16,~32, \ldots$

$1, 8, 27, 64, 125, \ldots$

$\dfrac{1}{2}$, $-\dfrac{1}{3}$, $\dfrac{1}{4}$, $-\dfrac{1}{5}$, $\dfrac{1}{6}, \ldots$

$1$, $-2$, $3$, $-4$, $5, \ldots$

$\dfrac{1}{2}, \dfrac{2}{3},~\dfrac{3}{4},~\dfrac{4}{5}, \ldots$

$\dfrac{1}{2}, \dfrac{4}{3},~\dfrac{9}{4},~\dfrac{16}{5},\ldots$

$1,1,2,6,24,120,720, \ldots$

$1,1,\dfrac{1}{2},\dfrac{1}{6},\dfrac{1}{24},\dfrac{1}{120},\ldots$

$\left\{\dfrac{3}{n}\right\}$

$\left\{\dfrac{-2}{n}\right\}$

$\left\{1-\dfrac{1}{n}\right\}$

$\left\{\dfrac{1}{n}+4\right\}$

$\left\{\dfrac{4n+2}{n}\right\}$

$\left\{ {\dfrac{2n+1}{n}}\right\}$

$\left\{ \left( \dfrac{2-n}{n^{2}}\right) ^{4}\right\}$

$\left\{ \left( \dfrac{n^{3}-2n}{n^{3}}\right) ^{2}\right\}$

$\left\{ \sqrt{\dfrac{n+1}{n^{2}}}\right\}$

$\left\{ \sqrt[3]{8-\dfrac{1}{n}}\right\}$

$\left\{ {\left( {1-\dfrac{1^{~}}{n}}\right) \left( {1-\dfrac{1^{~}}{n^{2}}}\right) }\right\}$

$\left\{ {\left( {1-\dfrac{1^{~}}{n}}\right) \left( {1-\dfrac{1^{~}}{n^{2}}}\right) \left( {1-\dfrac{1^{~}}{n^{3}}}\right) }\right\}$

$\left\{ \ln {\dfrac{n+1^{~}}{3n}}\right\}$

$\left\{\ln \dfrac{n^{2}+2}{2n^{2}+3}\right\}$

$\left\{e^{(4/n) -2}\right\}$

$\left\{e^{3+(6/n) }\right\}$

$\left\{\sin \dfrac{1}{n}\right\}$

$\left\{\cos \dfrac{1}{n}\right\}$

$\left\{ \dfrac{n^{2}-4}{n^{2}+n-2}\right\}$

$\left\{ \dfrac{n+2}{n^{2}+6n+8}\right\}$

$\left\{ {\dfrac{n^{2}}{2n+1}-\dfrac{n^{2}}{2n-1}}\right\}$

$\left\{ {\dfrac{6n^{4}-5}{7n^{4}+3}}\right\}$

$\left\{ {\dfrac{\sqrt{n}+2}{\sqrt{n}+5}}\right\}$

$\left\{ {\dfrac{\sqrt{n}}{e^{n}}}\right\}$

$\left\{ {\dfrac{n^{2}}{3^{n}}}\right\}$

$\left\{\dfrac{(n-1)^{2}}{e^{n}}\right\}$

$\left\{ \dfrac{(-1) ^{n}}{\sqrt{n}}\right\}$

$\left\{ \dfrac{\sin n}{n}\right\}$

$\left\{ \dfrac{\cos n}{n}\right\}$

$\left\{\cos \left( \pi n\right) \right\}$

$\left\{ \cos \left( \dfrac{\pi }{2}n\right) \right\}$

$\left\{ \sqrt{n}\right\}$

$\left\{ n^{2}\right\}$

$\left\{ \left( {\dfrac{1}{3}}\right) ^{n}\right\}$

$\left\{ \left( \dfrac{5}{4}\right)^{n}\right\}$

$\left\{ \left( \dfrac{\pi }{2}\right) ^{n}\right\}$

$\left\{ {\dfrac{n+(-1)^{n}}{n}}\right\}$

$\left\{ \dfrac{1}{n}+(-1)^{n}\right\}$

$\left\{ \dfrac{\ln n}{n}\right\}$

$\left\{\dfrac{\sin n}{n}\right\}$

$\left\{n+\dfrac{1}{n}\right\}$

$\left\{\dfrac{3}{n+1}\right\}$

$\left\{ \dfrac{n^{2}}{n+1}\right\}$

$\left\{ \dfrac{2^{n}}{n^{2}}\right\}$

$\left\{ \left( -\dfrac{1}{2}\right) ^{n}\right\}$

$\left\{ n^{1/2}\right\}$

$\left\{ \dfrac{3^{n}}{(n+1) ^{3}}\right\}$

$\left\{ {\dfrac{2n+1}{n}}\right\}$

$\left\{ \dfrac{\ln n}{\sqrt{n}}\right\}$

$\left\{ {\dfrac{\sqrt{n+1}}{n}}\right\}$

$\left\{ {\left( {\dfrac{1}{3}}\right) ^{n}}\right\}$

$\left\{ \dfrac{n^{2}}{5^{n}}\right\}$

$\left\{ {\dfrac{n!}{3^{n}}}\right\}$

$\left\{ {\dfrac{n!}{n^{2}}}\right\}$

$\left\{ ne^{-n}\right\}$

$\left\{\tan ^{-1}n\right\}$

$\left\{\dfrac{n}{n+1}\right\}$

$\left\{\dfrac{n}{n^{2}+1}\right\}$

$\left\{2-\dfrac{1}{n}\right\}$

$\left\{\dfrac{n}{2^{n}}\right\}$

$\left\{ {\dfrac{3}{n}+6}\right\}$

$\left\{ {2-\dfrac{4}{n}}\right\}$

$\left\{ \ln \left({\dfrac{n+1^{~}}{3n}}\right) \right\}$

$\left\{\cos\left( {{n\pi +\dfrac{\pi }{2}}}\right) \right\}$

$\left\{ (-1)^{n}\sqrt{n}\right\}$

$\left\{ {\dfrac{(-1)^{n}}{2n}}\right\}$

$\left\{ \dfrac{3^{n}+1}{4^{n}}\right\}$

$\left\{ {n+\sin \dfrac{1}{n}}\right\}$

$\left\{ {\dfrac{\ln (n+1)}{n+1}}\right\}$

$\left\{ \dfrac{\ln (n+1)}{\sqrt{n}}\right\}$

$\kern1.7pt\{0.5^{n}\}$

$\left\{(-2) ^{n}\right\}$

$\left\{\cos \dfrac{\pi}{n}\right\}$

$\left\{\sin \dfrac{\pi}{n}\right\}$

$\left\{\cos \left( \dfrac{n}{e^{n}}\right) \right\}$

$\left\{\sin \left( \dfrac{(n+1)^{3}}{e^{n}}\right) \right\}$

$\left\{e^{1/n}\right\}$

$\left\{ {\dfrac{1}{ne^{-n}}}\right\}$

$\left\{ 1+\left( \dfrac{1}{2}\right)^{n}\right\}$

$\left\{ 1-\left( \dfrac{1}{2}\right) ^{n}\right\}$

$\left\{ {\dfrac{n^{2}\tan ^{-1}n}{n^{2}+1}}\right\}$

$\left\{ n\sin \dfrac{1}{n}\right\}$

$\left\{ {\dfrac{n+\sin n}{n+\cos (4n) }}\right\}$

$\left\{ {\dfrac{n^{2}}{2n+1}\sin \dfrac{1}{n}}\right\}$

$\{\ln n-\ln (n+1)\}$

$\left\{ \ln n^{2}+\ln \dfrac{1}{n^{2}+1}\right\}$

$\left\{ {\dfrac{n^{2}}{\sqrt{n^{2}+1}}}\right\}$

$\left\{\dfrac{5^{n}}{(n+1)^{2}}\right\}$

$\left\{ \dfrac{2^{n}}{(2)(4)(6)\cdots (2n)}\right\}$

$\left\{ \dfrac{3^{n+1}}{(3)(6)(9)\cdots (3n)}\right\}$

The $n$th term of a sequence is $s_{n}=\dfrac{1}{n^{2}+n\cos n+1}.$ Does the sequence $\{ s_{n}\}$ converge or diverge? (Hint: Show that the derivative of $\dfrac{1}{x^{2}+x\cos x+1}$ is negative for $x>1$.)

Fibonacci Sequence The famous Fibonacci sequence $\left\{ u_{n}\right\}$ is defined recursively as $$u_{1}=1 \qquad u_{2}=1 \qquad u_{n+2}=u_{n}+u_{n+1} \quad n\geq 1$$

Stocking a Lake Mirror Lake is stocked with rainbow trout. Considering fish reproduction and natural death, along with vigorous efforts by fishermen to decimate the population, managers find that some ratio $r,$ $0<r<1,$ of the population persists from one stocking period to the next. If the lake is stocked with $h$ fish each year, the fish population $p_{n}$, in year $n$ of the stocking program, is approximately $p_{n}=rp_{n-1}+h.$ If $p_{0}$ is $3000,$ write a general expression for the $n$th term of the sequence in terms of $r$ and $h$ only. Does this sequence converge?

Electronics: A Discharging Capacitor A capacitor is an electronic device that stores an electrical charge. When connected across a resistor, it loses the charge (discharges) in such a way that during a fixed time interval, called the time constant, the charge stored in the capacitor is $\dfrac{1}{e}$ of the charge at the beginning of that interval.

Reflections in a Mirror A highly reflective mirror reflects $95\%$ of the light that falls on it. In a light box having walls made of this mirror, the light will reflect back-and-forth between the mirrors.

A Fission Chain Reaction A chain reaction is any sequence of events for which each event causes one or more additional events to occur. For example, in chain-reaction auto accidents, one car rear-ends another car, that car rear-ends another, and so on. In one type of nuclear fission chain reaction, a uranium-235 nucleus is struck by a neutron, causing it to break apart and release several more neutrons. Each of these neutrons strikes another nucleus, causing it to break apart and release additional neutrons, resulting in a chain reaction. In the fission of uranium-235 in nuclear reactors, each fission event releases an average of $2\dfrac{1}{2}$ neutrons, and each of these neutrons causes another fission event. The first fission is triggered by a single free neutron.

Show that if $0<r<1$, then $\lim\limits_{n\,\rightarrow \,\infty }\,r^{n}=0$. Hint: Let $r=\dfrac{1}{1+p}$, where $p>0.$ Then, by the Binomial Theorem, $$r^{n}=\dfrac{1}{(1+p)^{n}}= \dfrac{1}{1+np+n(n-1)\dfrac{p^{2}}{2}+\cdots +p^{n}}<\dfrac{1}{np}.$$

Use the result of Problem 132 to show that if $-1<r<0$, then $\lim\limits_{n\,\rightarrow \,\infty }\,r^{n}=0$.

Show that if $r>1$, then $\lim\limits_{n\rightarrow \,\infty }\,r^{n}=\infty$. [Hint: Let $r=1+p$, where $p>0$. Then by the Binomial Theorem, $r^{n}=(1+p)^{n}=1+np+n(n-1)\dfrac{p^{2}}{2}+\cdots+p^{n}>np$.]

Use the result of Problem 134 to show that if $r<-1$, then $\lim\limits_{n\,\rightarrow \,\infty }\,r^{n}$ does not exist. (Hint: $r^{n}$ oscillates between positive and negative values.)

Suppose $\{ s_{n}\}$ is a sequence of real numbers. Show that if $\lim\limits_{n\rightarrow \infty }s_{n}=L$ and if $f$ is a function that is continuous at $L$ and is defined for all numbers $s_{n},$ then $\lim\limits_{n\rightarrow \infty }f( s_{n})=f(L) .$

Show that if $\lim\limits_{n\,\rightarrow \,\infty}\,s_{n}=L$, then $\lim\limits_{n\rightarrow \,\infty }\,\vert s_{n}\vert$ exists and $\lim\limits_{n\,\rightarrow \,\infty }\,\vert s_{n}\vert =\left\vert L\right\vert$. Is the converse true?

The Limit of a Sequence Is Unique Show that a convergent sequence $\{ s_{n}\}$ cannot have two distinct limits.

Review the definition of the limit at infinity of a function from Section 1.6. Write a paragraph that compares and contrasts the limit at infinity of a function $f$ and the limit of a sequence $\{ s_{n}\}$.

Let $a_{1}>$ $0$ and $b_{1}>0$ be two real numbers for which $a_{1}>b_{1}$. Define sequences $\{a_{n}\}$ and $\{b_{n}\}$ as $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \qquad b_{n+1}=\sqrt{a_{n}b_{n}}$$

$s_{n}=\dfrac{2^{n-1}\cdot 4^{n}}{n!}$

$s_{n}=\dfrac{n!}{3^{n}\cdot 4^{n}}$

$s_{n}=\dfrac{n!}{3^{n}+8n}$

Show that $\left\{( 3^{n}+5^{n}) ^{1/n}\right\}$ converges.

Let $N$ be a fixed positive number and define a sequence by $\{a_{n+1}\} =\left\{ \dfrac{1}{2}\left[ a_{n}+\dfrac{N}{a_{n}}\right] \right\}$, where $a_{1}$ is a positive number.

Show that $\{ s_{n}\} =\left\{ \dfrac{1\cdot3\cdot 5\cdot \cdots \cdot \left( 2n-1\right) }{2\cdot 4\cdot 6\cdot \cdots\cdot 2n}\right\}$ is bounded and monotonic.

Show that $\{ s_{n}\} =\left\{ \left( 1+\dfrac{1}{n}\right) ^{n}\right\}$ is increasing and bounded from above. Hint: Use the Binomial Theorem to expand $\left( 1+\dfrac{1}{n}\right) ^{n}.$

Let $\{ s_{n}\}$ be a convergent sequence, and suppose the $n$th term of the sequence $\{a_{n}\}$ is the arithmetic mean (average) of the first $n$ terms of $\{ s_{n}\}$. That is, $a_{n}=\dfrac{1}{n}\left[ s_{1}+s_{2}+\cdots +s_{n}\right] .$ Show that $\{a_{n}\}$ converges and has the same limit as $\{ s_{n}\}.$

Area Let $A_{n}$ be the area enclosed by a regular $n$-sided polygon inscribed in a circle of radius $R$. Show that:

Area Let $A_{n}$ be the area enclosed by a regular $n$-sided polygon circumscribed around a circle of radius $r$. Show that:

Perimeter Suppose $P_{n}$ is the perimeter of a regular $n$-sided polygon inscribed in a circle of radius $R$. Show that:

Cauchy Sequence A sequence $\{ s_{n}\}$ is said to be a Cauchy sequence if and only if for each $\varepsilon >0$, there exists a positive integer $N$ for which $$\left\vert s_{n}-s_{m}\right\vert < \varepsilon\qquad \hbox{ for all }n, m>N$$

Show that every convergent sequence is a Cauchy sequence.