Section 1
Sequences
(a) What is a sequence?(b) What does it mean to say that $\lim _{h \rightarrow \infty} a_{h}=8 ?$(c) What does it mean to say that $|$ im. $\ldots . a_{\infty}=x ?$
(a) What is a convergent sequence? Give two examples.(b) What is a divergent sequence? Give two examples.
List the first six terms of the sequence defined by$$a_{n}=\frac{n}{2 n+1}$$Does the sequence appear to have a limit? If so, find it.
List the first nine terms of the sequence $\{\cos (n \pi / 3)\} .$ Does this sequence appear to have a limit? If so, find it. If not, explain why.
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\left(1, \frac{1}{3}, 1,3,-, \ldots\right)$$
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\left(1,1,1, \frac{1}{27}, \frac{1}{61}, \ldots\right)$$
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\{2,7,12,17, \dots\}$$
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\left(-\frac{1}{4}, \frac{2}{\square},-\frac{3}{16}, \frac{4}{25}, \ldots\right)$$
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\{1,-\frac{2}{3}, \quad ,\}$$
Find a formula for the general term $a$, of the sequence. assuming that the pattern of the first few terms continues.$$\{5,1,5,1,5,1$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{3+5 n^{2}}{n+n^{2}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{n^{3}}{n^{3}+1}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=1-(0.2)^{n}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{n^{3}}{n+1}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=e^{1 / n}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{3^{n+2}}{5^{n}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\tan \left(\frac{2 n \pi}{1+8 n}\right)$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\sqrt{\frac{n+1}{9 n+1}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{(-1)^{n-1} n}{n^{2}+1}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{(-1)^{n} n^{3}}{n^{3}+2 n^{2}+1}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$\left\{\frac{e^{n}+e^{-n}}{e^{2 n}-1}\right\}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\cos (2 / n)$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$\left(n^{2} e^{-x}\right)$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$\{\arctan 2 n\}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{\cos ^{2} n}{2^{n}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$(n \cos n \pi)$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\left(1+\frac{2}{n}\right)^{n}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\sqrt[*]{2^{1+3 n}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$\left\{\frac{(2 n-1) !}{(2 n+1) !}\right\}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{\sin 2 n}{1+\sqrt{n}}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$\{0,1,0,0,1,0,0,0,1, \ldots 1\}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\frac{(\ln n)^{2}}{n}$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{n}=\ln \left(2 n^{2}+1\right)-\ln \left(n^{2}+1\right)$$
Determine whether the sequence converges or diverges. If it converges, find the limit.$$a_{a}=\frac{(-3)^{c}}{n !}$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{n}=1+(-2 / e)^{n}$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{n}=\sqrt{n} \sin (\pi / \sqrt{n})$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{n}=\sqrt{\frac{3+2 n^{2}}{8 n^{2}+n}}$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{n}=\sqrt[4]{3^{n}+5^{n}}$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{n}=\frac{n^{2} \cos n}{1+n^{2}}$$
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 559 for advice on graphing sequences.)$$a_{x}=\frac{1 \cdot 3 \cdot 5 \cdot \quad \cdot(2 n-1)}{(2 n)^{n}}$$
If $\$ 1000$ is invested at $6 \%$ interest. compounded annually. then after $n$ years the investment is worth $a_{x}=1000(1.06)^{x}$ dollars.(a) Find the first five terms of the sequence $\left\{a_{n}\right\}$(b) Is the sequence convergent or divergent? Explain.
If you deposit $\$ 100$ at the end of every month into an account that pays $3 \%$ interest per year compounded monthly, the amount of interest accumulated after $n$ months is given by the sequence$$l_{n}=100\left(\frac{1.0025^{n}-1}{0.0025}-n\right)$$(a) Find the first six terms of the sequence.(b) How much interest will you have earned after two years?
A fish farmer has 5000 catfish in his pond. The number of calfish increases by 89 per month and the farmer harvests 300 catfish per month.(a) Show that the catfish population $P$, after $n$ months is given recursively by$$R_{n}=1.08 P_{n-1}-300 \quad P_{0}=5000$$(b) How many catfish are in the pond after six months?
Find the first 40 terms of the sequence defined by$$a_{n+1}=\left\{\begin{array}{ll}\frac{1}{2} a_{n} & \text { if } a_{n} \text { is an even number } \\3 a_{n}+1 & \text { if } a_{n} \text { is an odd number }\end{array}\right.$$and $a_{1}=11 .$ Do the same if $a_{1}=25 .$ Make a conjecture about this type of sequence.
(a) Determine whether the sequence defined as follows is convergent or divergent:$$a_{1}=1 \quad a_{n+1}=4-a_{n} \quad \text { for } n \geq 1$$(b) What happens if the first term is $a_{1}=2 ?$
(a) If $\lim _{a \rightarrow \infty} a_{n}=L,$ what is the value of $\lim _{n \rightarrow \infty} a_{n+1}$ ?(b) A sequence $\left\{a_{n}\right\}$ is defined by$$a_{1}=1 \quad a_{n+1}=1 /\left(1+a_{2}\right) \text { for } n \geq 1$$Find the first ten terms of the sequence correct to five decimal places. Does it appear that the sequence is convergent? If so, estimate the value of the limit to three decimal places.(c) Assuming that the sequence in part (b) has a limit, use part (a) to find its exact value. Compare with your estimate from part (b).
(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the $n$ th month? Show that the answer is $f_{n},$ where $\left\{f_{0}\right\}$ is the Fibonacci sequence defined in Example $3(\mathrm{c})$(b) Let $a_{\alpha}=f_{\alpha+1} / f_{\alpha}$ and show that $a_{\alpha-1}=1+1 / a_{\alpha-2}$Assuming that $\left\{a_{n}\right\}$ is convergent, find its limit.
Find the limit of the sequence$$(\sqrt{2}, \sqrt{2 \sqrt{2}}, \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots)$$
Determine whether the sequence is increasing, decreasing. or not monotonic. Is the sequence bounded?$$a_{a}=\frac{1}{2 n+3}$$
Determine whether the sequence is increasing, decreasing. or not monotonic. Is the sequence bounded?$$a_{n}=\frac{2 n-3}{2}$$
Determine whether the sequence is increasing, decreasing. or not monotonic. Is the sequence bounded?$$a_{n}=n(-1)^{n}$$
Determine whether the sequence is increasing, decreasing. or not monotonic. Is the sequence bounded?$$a_{n}=n+\frac{1}{n}$$
Suppose you know that $\left\{a_{n}\right\}$ is a decreasing sequence and all its terms lie between the numbers 5 and $8 .$ Explain why the sequence has a limit. What can you say about the value of the limit?
A sequence $\left\{a, 1 \text { is given by } a_{1}=\sqrt{2}, a_{n+1}=\sqrt{2+a_{n}}\right.$(a) By induction or otherwise, show that $\left\{a_{a}\right\}$ is increasing and bounded above by $3 .$ Apply the Monotonic Sequence Theorem to show that lim, $\rightarrow a_{n}$ exists.(b) Find $\lim _{x \rightarrow \infty}$
Show that the sequence defined by$$a_{1}=1 \quad a_{n+1}=3-\frac{1}{a_{n}}$$is increasing and $a_{n}<3$ for all $n$. Deduce that $\left\{a_{0}\right\}$ is convergent and find its limit.
Show that the sequence defined by$a_{1}=2 \quad a_{n+1}=\frac{1}{3-a_{n}}$satisfies $0<a_{a} \leqslant 2$ and is decreasing. Deduce that the sequence is convergent and find its limit.
We know that $\lim _{a \rightarrow \infty}(0.8)^{a}=0$ [from (7) with $r=0.81$. Use logarithms to determine how large $n$ has to be so that $(0.8)^{x}<0.000001$
(a) Let $a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a))$$a_{n+1}=f\left(a_{n}\right),$ where $f$ is a continuous function. If $\lim _{x \rightarrow \infty} a_{n}=L,$ show that $f(L)-L$(b) Illustrate part (a) by taking $f(x)=\cos x, a=1,$ and estimating the value of $L$ to five decimal places.
The size of an undisturbed fish population has been modeled by the formula$$p_{n+1}=\frac{b p_{n}}{a+p_{n}}$$where $p_{n}$ is the fish population after $n$ years and $a$ and $b$ are positive constants that depend on the species and its environment. Suppose that the population in year 0 is $p_{0}>0$(a) Show that if $\left\{p_{n}\right\}$ is convergent, then the only possible values for its limit are 0 and $b-a$(b) Show that $p_{x+1}<(b / a) p_{x}$(c) Use part (b) to show that if $a>b$, then $\lim _{x \rightarrow \infty} p_{x}=0$ :in other words, the population dies out.(d) Now assume that $a<b .$ Show that if $p_{0}<b-a$, then $\left\{p_{n}\right\}$ is increasing and $0<p_{n}<b-a .$ Show also thatif $p_{0}>b-a,$ then $\left\langle p_{0}\right\}$ is decreasing and $p_{n}>b-a$ Deduce that if $a<b,$ then $\lim _{x \rightarrow \infty} p_{x}=b-a$
A sequence is defined recursively by$$a_{1}=1 \quad a_{n+1}=1+\frac{1}{1+a_{n}}$$Find the first eight terms of the sequence $\left\{a_{a}\right\} .$ What do you notice about the odd terms and the even terms? By considering the odd and even terms separately, show that $\left\langle a_{n}\right\rangle$ is convergent and deduce that$$\lim _{n \rightarrow \infty} a_{n}=\sqrt{2}$$This gives the continued fraction expansion$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+}}$$