Question

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

Question

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

Question

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.

Question

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.]

Question

True or False A bounded sequence is convergent.

Question

True or False An unbounded sequence is divergent.

Question

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

Question

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

Question

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 \(n\geq 1\).

Question

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.]

Question

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

Question

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

Question

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

Question

\(s_{n}=\dfrac{2}{n^{2}}\)

Question

\(s_{n}=\ln n\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(s_{n}=\dfrac{n!}{2^{n}}\)

Question

\(s_{n}=\dfrac{n!}{n^{2}}\)

Question

\(2, 4, 6, 8, 10, \ldots\)

Question

\(1, 3, 5, 7, 9, \ldots\)

Question

\(2,~4,~8,~16,~32, \ldots\)

Question

\(1, 8, 27, 64, 125, \ldots\)

Question

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

Question

\(1\), \(-2\), \(3\), \(-4\), \(5, \ldots\)

Question

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

Question

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

Question

\(1,1,2,6,24,120,720, \ldots\)

Question

\(1,1,\dfrac{1}{2},\dfrac{1}{6},\dfrac{1}{24},\dfrac{1}{120},\ldots\)

Question

\(\left\{\dfrac{3}{n}\right\}\)

Question

\(\left\{\dfrac{-2}{n}\right\}\)

Question

\(\left\{1-\dfrac{1}{n}\right\}\)

Question

\(\left\{\dfrac{1}{n}+4\right\}\)

Question

\(\left\{\dfrac{4n+2}{n}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{\sin \dfrac{1}{n}\right\}\)

Question

\(\left\{\cos \dfrac{1}{n}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ {\dfrac{\sqrt{n}+2}{\sqrt{n}+5}}\right\}\)

Question

\(\left\{ {\dfrac{\sqrt{n}}{e^{n}}}\right\}\)

Question

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

Question

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

Question

Question

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

Question

\(\left\{ \dfrac{\sin n}{n}\right\}\)

Question

\(\left\{ \dfrac{\cos n}{n}\right\}\)

Question

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

Question

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

Question

\(\left\{ \sqrt{n}\right\}\)

Question

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

Question

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ \dfrac{\ln n}{n}\right\}\)

Question

\(\left\{\dfrac{\sin n}{n}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ n^{1/2}\right\}\)

Question

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

Question

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

Question

\(\left\{ \dfrac{\ln n}{\sqrt{n}}\right\}\)

Question

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

Question

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

Question

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

Question

\(\left\{ {\dfrac{n!}{3^{n}}}\right\}\)

Question

\(\left\{ {\dfrac{n!}{n^{2}}}\right\}\)

Question

\(\left\{ ne^{-n}\right\}\)

Question

\(\left\{\tan ^{-1}n\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ {\dfrac{3}{n}+6}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ {n+\sin \dfrac{1}{n}}\right\}\)

Question

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

Question

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

Question

\(\kern1.7pt\{0.5^{n}\}\)

Question

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

Question

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

Question

\(\left\{\sin \dfrac{\pi}{n}\right\}\)

Question

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

Question

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

Question

\(\left\{e^{1/n}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

\(\left\{ n\sin \dfrac{1}{n}\right\}\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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 \(x>1\).)

Question

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

1. Write the first eight terms of the Fibonacci sequence.
2. Verify that the \(n\)th term is given by \[ u_{n}=\frac{(1+\sqrt{5})^{n}-(1-\sqrt{5})^{n}}{2^{n}\sqrt{5}} \] (Hint: Show that \(u_{1}=1,~u_{2}=1\), and \(u_{n+2}=u_{n+1}+u_{n}\).)

Question

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?

Question

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.

1. Develop a sequence for the charge remaining after \(n\) time constants if the initial charge is \(Q_{0}.\)
2. Does this sequence converge? If yes, to what?

Question

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.

1. If the original intensity of the light is \(I_{0}\) before it falls on a mirror, develop a sequence to describe the intensity of the light after \(n\) reflections.
2. How many reflections are needed to reduce the light intensity by at least \(98\%\)?

Question

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.

1. Develop a sequence for the average number of neutrons, that is, fission events, that occur at the \(n\)th event if we start with one such event.
2. Does the sequence converge or diverge?
3. Interpret the answer found in (b).

Question

1. Show that the sequence \(\{ e^{n/(n+2)}\}\) converges.
2. Find \(\lim\limits_{n\rightarrow \infty }e^{n/(n+2) }\).
3. Graph \(y=e^{n/(n+2) }\). Does the graph confirm the results of (a) and (b)?

Question

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, \[ r^{n}=\dfrac{1}{(1+p)^{n}}= \dfrac{1}{1+np+n(n-1)\dfrac{p^{2}}{2}+\cdots +p^{n}}<\dfrac{1}{np}. \]

Question

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

Question

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, \(r^{n}=(1+p)^{n}=1+np+n(n-1)\dfrac{p^{2}}{2}+\cdots+p^{n}>np\).]

Question

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.)

Question

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 \(\lim\limits_{n\rightarrow \infty }f( s_{n})=f(L) .\)

Question

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?

Question

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

Question

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 \(\{ s_{n}\}\).

Question

1. Show that the sequence \(\{ \ln n\}\) is increasing.
2. Show that the sequence \(\{ \ln n\}\) is unbounded from above.
3. Conclude \(\{ \ln n\}\) diverges.
4. Find the smallest number \(N\) so that \(\ln N>20.\)
5. Graph \(y=\ln x\) and zoom in for \(x\) large.
6. Does the graph confirm the result in (c)?

Question

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 \[ a_{n+1}=\frac{a_{n}+b_{n}}{2}, \qquad b_{n+1}=\sqrt{a_{n}b_{n}} \]

1. Show that \(b_{n}<b_{n+1}<a_{1}\) for all \(n\).
2. Show that \(b_{1}<a_{n+1}<a_{n}\) for all \(n\).

   553

3. Show that \(0<a_{n+1}-b_{n+1}<\dfrac{a_{1}-b_{1}}{2^{n}}\).
4. Show that \(\lim\limits_{n\,\rightarrow \,\infty }a_{n}\) and \(\lim\limits_{n\,\rightarrow \,\infty }\,b_{n}\) each exist and are equal.

Question

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

Question

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

Question

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

Question

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

Question

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.

1. Show that the sequence \(\{a_{n}\}\) converges to \(\sqrt{N}\).
2. Use this sequence to approximate \(\sqrt{28}\) rounded to three decimal places. How accurate is \(a_{3}\)? \(a_{6}\)?

Question

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.

Question

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

Question

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 \(\{ s_{n}\}.\)

Question

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

1. \(A_{n}=\dfrac{n}{2}R^{2}\sin\! \left( \dfrac{2\pi }{n}\right)\).
2. \(\lim\limits_{n\rightarrow \infty}A_{n}=\lim\limits_{n\rightarrow \infty }\left[ \dfrac{n}{2}R^{2}\sin\! \left( \dfrac{2\pi }{n}\right) \right] =\pi R^{2}\) (the area of a circle of radius \(R\)).

Question

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

1. \(A_{n}=nr^{2}\tan\! \left( \dfrac{\pi }{n}\right)\).
2. \(\lim\limits_{n\rightarrow \infty }A_{n}=\pi r^{2}\) (the area of a circle of radius \(r\)). (Hint: \(r,\) called the apothem of the polygon, is the perpendicular distance from the center of the polygon to the midpoint of a side.)

Question

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

1. \(P_{n}=2nR\sin \!\left( \dfrac{\pi }{n}\right)\).
2. \({\lim\limits_{n\rightarrow \infty }P_{n}=2\pi R}\) (the circumference of a circle of radius \(R\)).

Question

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.