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Chapter 8

SERIES

Educators


Problem 1

(a) What is a sequence?
(b) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=8 ?$
(c) What does it mean to say that $\lim _{n \rightarrow \infty} a_{n}=\infty$ ?

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Problem 2

(a) What is a convergent sequence? Give two examples.
(b) What is a divergent sequence? Give two examples.

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Problem 3

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.

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Problem 4

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.

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Problem 5

$5-8=$ Find a formula for the general term $a_{n}$ of the sequence,
assuming that the pattern of the first few terms continues.
$$\left\{-3,2,-\frac{4}{3}, \frac{8}{9},-\frac{16}{27}, \ldots\right\}$$

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Problem 6

$5-8=$ Find a formula for the general term $a_{n}$ of the sequence,
assuming that the pattern of the first few terms continues.
$$\left\{1,-\frac{1}{3}, \frac{1}{9},-\frac{1}{27}, \frac{1}{81}, \ldots\right\}$$

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Problem 7

$5-8=$ Find a formula for the general term $a_{n}$ of the sequence,
assuming that the pattern of the first few terms continues.
$$\left\{\frac{1}{2},-\frac{4}{3}, \frac{9}{4},-\frac{16}{5}, \frac{25}{6}, \ldots\right\}$$

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Problem 8

$5-8=$ Find a formula for the general term $a_{n}$ of the sequence,
assuming that the pattern of the first few terms continues.
$$\{5,8,11,14,17, \ldots\}$$

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Problem 9

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=1-(0.2)^{n}$$

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Problem 10

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{n^{3}}{n^{3}+1}$$

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Problem 11

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{3+5 n^{2}}{n+n^{2}}$$

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Problem 12

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{n^{3}}{n+1}$$

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Problem 13

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\tan \left(\frac{2 n \pi}{1+8 n}\right)$$

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Problem 14

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{3^{n+2}}{5^{n}}$$

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Problem 15

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{n^{2}}{\sqrt{n^{3}+4 n}}$$

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Problem 16

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\sqrt{\frac{n+1}{9 n+1}}$$

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Problem 17

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{(-1)^{n}}{2 \sqrt{n}}$$

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Problem 18

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{(-1)^{n+1} n}{n+\sqrt{n}}$$

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Problem 19

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\cos (n / 2)$$

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Problem 20

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\cos (2 / n)$$

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Problem 21

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$\left\{\frac{(2 n-1) !}{(2 n+1) !}\right\}$$

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Problem 22

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{\tan ^{-1} n}{n}$$

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Problem 23

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$\left\{n^{2} e^{-n}\right\}$$

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Problem 24

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\ln (n+1)-\ln n$$

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Problem 25

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{\cos ^{2} n}{2^{n}}$$

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Problem 26

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=2^{-n} \cos n \pi$$

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Problem 27

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\left(1+\frac{2}{n}\right)^{n}$$

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Problem 28

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{\sin 2 n}{1+\sqrt{n}}$$

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Problem 29

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$\{0,1,0,0,1,0,0,0,1, \ldots\}$$

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Problem 30

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{(\ln n)^{2}}{n}$$

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Problem 31

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\ln \left(2 n^{2}+1\right)-\ln \left(n^{2}+1\right)$$

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Problem 32

$9-32$ n Determine whether the sequence converges or diverges.
If it converges, find the limit.
$$a_{n}=\frac{(-3)^{n}}{n !}$$

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Problem 33

If $\$ 1000$ is invested at 6$\%$ interest, compounded annually,
then after $n$ years the investment is worth $a_{n}=1000(1.06)^{n}$
dollars.
(a) Find the first five terms of the sequence $\left\{a_{n}\right\} .$
(b) Is the sequence convergent or divergent? Explain.

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Problem 34

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.

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Problem 35

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?

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Problem 36

(a) If $\left\{a_{n}\right\}$ is convergent, show that
$$\lim _{n \rightarrow \infty} a_{n+1}=\lim _{n \rightarrow \infty} a_{n}$$
(b) A sequence $\left\{a_{n}\right\}$ is defined by $a_{1}=1$ and
$a_{n+1}=1 /\left(1+a_{n}\right)$ for $n \geqslant 1 .$ Assuming that $\left\{a_{n}\right\}$ is
convergent, find its limit.

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Problem 37

$37-40=$ Determine whether the sequence is increasing,
decreasing, or not monotonic. Is the sequence bounded?
$$a_{n}=\frac{1}{2 n+3}$$

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Problem 38

$37-40=$ Determine whether the sequence is increasing,
decreasing, or not monotonic. Is the sequence bounded?
$$a_{n}=\frac{2 n-3}{3 n+4}$$

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Problem 39

$37-40=$ Determine whether the sequence is increasing,
decreasing, or not monotonic. Is the sequence bounded?
$$a_{n}=n(-1)^{n}$$

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Problem 40

$37-40=$ Determine whether the sequence is increasing,
decreasing, or not monotonic. Is the sequence bounded?
$$a_{n}=n+\frac{1}{n}$$

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Problem 41

Find the limit of the sequence
$$\{\sqrt{2}, \sqrt{2 \sqrt{2}}, \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots\}$$

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Problem 42

A sequence $\left\{a_{n}\right\}$ is given by $a_{1}=\sqrt{2}, a_{n+1}=\sqrt{2+a_{n}}$
(a) By induction or otherwise, show that $\left\{a_{n}\right\}$ is increasing
and bounded above by $3 .$ Apply the Monotonic
Sequence Theorem to show that lim $_{n \rightarrow \infty} a_{n}$ exists.
(b) Find $\lim _{n \rightarrow \infty} a_{n} .$

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Problem 43

Use induction to show that the sequence defined by $a_{1}=1$
$a_{n+1}=3-1 / a_{n}$ is increasing and $a_{n}<3$ for all $n .$ Deduce
that $\left\{a_{n}\right\}$ is convergent and find its limit.

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Problem 44

Show that the sequence defined by
$$a_{1}=2 \quad a_{n+1}=\frac{1}{3-a_{n}}$$
satisfies $0 < a_{n} \leqslant 2$ and is decreasing. Deduce that the
sequence is convergent and find its limit.

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Problem 45

(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 $\left\{f_{n}\right\}$ is the Fibonacci
sequence defined in Example 3$(\mathrm{c}) .$
(b) Let $a_{n}=f_{n+1} / f_{n}$ and show that $a_{n-1}=1+1 / a_{n-2}$
Assuming that $\left\{a_{n}\right\}$ is convergent, find its limit.

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Problem 46

(a) Let $a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldots$
$a_{n+1}=f\left(a_{n}\right),$ where $f$ is a continuous function. If
$\lim _{n \rightarrow \infty} a_{n}=L,$ 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.

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Problem 47

We know that $\lim _{n \rightarrow \infty}(0.8)^{n}=0[$ from $[8]$ with $r=0.8]$
Use logarithms to determine how large $n$ has to be so that
$(0.8)^{n}<0.000001 .$

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Problem 48

Use Definition 2 directly to prove that lim_{n} \rightarrow x ^ { n } $=0$
when $|r| <1 $

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Problem 49

Prove Theorem $6 .$
[Hint: Use either Definition 2 or the Squeeze Theorem.]

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Problem 50

Prove the Continuity and Convergence Theorem.

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Problem 51

Prove that if $\lim _{n \rightarrow \infty} a_{n}=0$ and $\left\{b_{n}\right\}$ is bounded, then
$\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=0$

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Problem 52

(a) Show that if $\lim _{n \rightarrow \infty} a_{2 n}=L$ and $\lim _{n \rightarrow \infty} a_{2 n+1}=L$
then $\left\{a_{n}\right\}$ is convergent and $\lim _{n \rightarrow \infty} a_{n}=L$
(b) If $a_{1}=1$ and
$$a_{n+1}=1+\frac{1}{1+a_{n}}$$
find the first eight terms of the sequence $\left\{a_{n}\right\} .$ Then use
part (a) to show that lim $_{n \rightarrow \infty} a_{n}=\sqrt{2} .$ This gives the
continued fraction expansion
$$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\cdots}}$$

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Problem 53

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 $p_{n+1}<(b / a) p_{n} .$
(c) Use part (b) to show that if $a > b,$ then $\lim _{n \rightarrow \infty} p_{n}=0$
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 that
if $p_{0} > b-a,$ then $\left\{p_{n}\right\}$ is decreasing and $p_{n} > b-a$
Deduce that if $a < b,$ then $\lim _{n \rightarrow \infty} p_{n}=b-a$

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