## 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. Check back soon! 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. Check back soon! 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? Check back soon! 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. Check back soon! Problem 37$37-40=$Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{1}{2 n+3}$$ Check back soon! Problem 38$37-40=$Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=\frac{2 n-3}{3 n+4}$$ Check back soon! Problem 39$37-40=$Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=n(-1)^{n}$$ Check back soon! Problem 40$37-40=$Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? $$a_{n}=n+\frac{1}{n}$$ Check back soon! Problem 41 Find the limit of the sequence $$\{\sqrt{2}, \sqrt{2 \sqrt{2}}, \sqrt{2 \sqrt{2 \sqrt{2}}}, \ldots\}$$ Check back soon! 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$\lim _{n \rightarrow \infty} a_{n} .$Check back soon! Problem 43 Use induction to show that the sequence defined by$a_{1}=1a_{n+1}=3-1 / a_{n}$is increasing and$a_{n}<3$for all$n .$Deduce that$\left\{a_{n}\right\}$is convergent and find its limit. Check back soon! 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. Check back soon! 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. Check back soon! Problem 46 (a) Let$a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldotsa_{n+1}=f\left(a_{n}\right),$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. Check back soon! 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$(0.8)^{n}<0.000001 .$Check back soon! Problem 48 Use Definition 2 directly to prove that lim_{n} \rightarrow x ^ { n }$=0$when$|r| <1 $Check back soon! Problem 49 Prove Theorem$6 .$[Hint: Use either Definition 2 or the Squeeze Theorem.] Check back soon! Problem 50 Prove the Continuity and Convergence Theorem. Check back soon! Problem 51 Prove that if$\lim _{n \rightarrow \infty} a_{n}=0$and$\left\{b_{n}\right\}$is bounded, then$\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=0$Check back soon! 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 $$\sqrt{2}=1+\frac{1}{2+\frac{1}{2+\cdots}}$$ Check back soon! 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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