Problem 1

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=\frac{1-n}{n^{2}}

$$

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Problem 2

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=\frac{1}{n !}

$$

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Problem 3

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=\frac{(-1)^{n+1}}{2 n-1}

$$

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Problem 4

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=2+(-1)^{n}

$$

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Problem 5

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=\frac{2^{n}}{2^{n+1}}

$$

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Problem 6

Each of Exercises $1-6$ gives a formula for the $n$ th term $a_{n}$ of a sequence $\left\{a_{n}\right\} .$ Find the values of $a_{1}, a_{2}, a_{3},$ and $a_{4}$ .

$$

a_{n}=\frac{2^{n}-1}{2^{n}}

$$

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Problem 7

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=1, \quad a_{n+1}=a_{n}+\left(1 / 2^{n}\right)

$$

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Problem 8

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=1, \quad a_{n+1}=a_{n} /(n+1)

$$

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Problem 9

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=2, \quad a_{n+1}=(-1)^{n+1} a_{n} / 2

$$

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Problem 10

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=-2, \quad a_{n+1}=n a_{n} /(n+1)

$$

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Problem 11

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=a_{2}=1, \quad a_{n+2}=a_{n+1}+a_{n}

$$

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Problem 12

Each of Exercises $7-12$ gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.

$$

a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n}

$$

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Problem 13

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $1,-1,1,-1,1, \ldots$

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Problem 14

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $-1,1,-1,1,-1, \ldots$

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Problem 15

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $1,-4,9,-16,25, \dots$

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Problem 16

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \dots$

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Problem 17

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $0,3,8,15,24, \dots$

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Problem 18

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $-3,-2,-1,0,1, \ldots$

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Problem 19

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $1,5,9,13,17, \ldots$

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Problem 20

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $2,6,10,14,18, \ldots$

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Problem 21

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $1,0,1,0,1, \ldots$

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Problem 22

In Exercises $13-22,$ find a formula for the $n$ th term of the sequence.

The sequence $0,1,1,2,2,3,3,4, \ldots$

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Problem 23

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=2+(0.1)^{n}

$$

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Problem 24

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n+(-1)^{n}}{n}

$$

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Problem 25

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1-2 n}{1+2 n}

$$

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Problem 26

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{2 n+1}{1-3 \sqrt{n}}

$$

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Problem 27

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1-5 n^{4}}{n^{4}+8 n^{3}}

$$

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Problem 28

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n+3}{n^{2}+5 n+6}

$$

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Problem 29

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n^{2}-2 n+1}{n-1}

$$

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Problem 30

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1-n^{3}}{70-4 n^{2}}

$$

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Problem 31

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=1+(-1)^{n}

$$

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Problem 32

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=(-1)^{n}\left(1-\frac{1}{n}\right)

$$

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Problem 33

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{n+1}{2 n}\right)\left(1-\frac{1}{n}\right)

$$

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Problem 34

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(2-\frac{1}{2^{n}}\right)\left(3+\frac{1}{2^{n}}\right)

$$

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Problem 35

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{(-1)^{n+1}}{2 n-1}

$$

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Problem 36

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(-\frac{1}{2}\right)^{n}

$$

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Problem 37

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt{\frac{2 n}{n+1}}

$$

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Problem 38

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1}{(0.9)^{n}}

$$

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Problem 39

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sin \left(\frac{\pi}{2}+\frac{1}{n}\right)

$$

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Problem 40

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=n \pi \cos (n \pi)

$$

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Problem 41

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{\sin n}{n}

$$

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Problem 42

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{\sin ^{2} n}{2^{n}}

$$

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Problem 43

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n}{2^{n}}

$$

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Problem 44

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{3^{n}}{n^{3}}

$$

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Problem 45

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{\ln (n+1)}{\sqrt{n}}

$$

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Problem 46

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{\ln n}{\ln 2 n}

$$

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Problem 47

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=8^{1 / n}

$$

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Problem 48

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=(0.03)^{1 / n}

$$

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Problem 49

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(1+\frac{7}{n}\right)^{n}

$$

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Problem 50

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(1-\frac{1}{n}\right)^{n}

$$

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Problem 51

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt[n]{10 n}

$$

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Problem 52

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt[n]{n^{2}}

$$

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Problem 53

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{3}{n}\right)^{1 / n}

$$

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Problem 54

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=(n+4)^{1 /(n+4)}

$$

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Problem 55

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{\ln n}{n^{1 / n}}

$$

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Problem 56

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

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

$$

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Problem 57

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt[n]{4^{n} n}

$$

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Problem 58

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt[n]{3^{2 n+1}}

$$

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Problem 59

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n !}{n^{n}} \quad(\text {Hint} : \text { Compare with } 1 / n .)

$$

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Problem 60

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{(-4)^{n}}{n !}

$$

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Problem 61

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n !}{10^{6 n}}

$$

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Problem 62

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n !}{2^{n} \cdot 3^{n}}

$$

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Problem 63

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{1}{n}\right)^{1 /(\ln n)}

$$

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Problem 64

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\ln \left(1+\frac{1}{n}\right)^{n}

$$

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Problem 65

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{3 n+1}{3 n-1}\right)^{n}

$$

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Problem 66

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{n}{n+1}\right)^{n}

$$

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Problem 67

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{x^{n}}{2 n+1}\right)^{1 / n}, \quad x>0

$$

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Problem 68

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(1-\frac{1}{n^{2}}\right)^{n}

$$

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Problem 69

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{3^{n} \cdot 6^{n}}{2^{-n} \cdot n !}

$$

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Problem 70

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{(10 / 11)^{n}}{(9 / 10)^{n}+(11 / 12)^{n}}

$$

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Problem 71

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\tanh n

$$

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Problem 72

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sinh (\ln n)

$$

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Problem 73

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{n^{2}}{2 n-1} \sin \frac{1}{n}

$$

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Problem 74

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=n\left(1-\cos \frac{1}{n}\right)

$$

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Problem 75

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\tan ^{-1} n

$$

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Problem 76

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1}{\sqrt{n}} \tan ^{-1} n

$$

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Problem 77

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}}

$$

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Problem 78

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\sqrt[n]{n^{2}+n}

$$

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Problem 79

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{(\ln n)^{200}}{n}

$$

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Problem 80

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{(\ln n)^{5}}{\sqrt{n}}

$$

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Problem 81

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=n-\sqrt{n^{2}-n}

$$

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Problem 82

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1}{\sqrt{n^{2}-1}-\sqrt{n^{2}+n}}

$$

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Problem 83

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\frac{1}{n} \int_{1}^{n} \frac{1}{x} d x

$$

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Problem 84

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $23-84$ converge, and which diverge? Find the limit of each convergent sequence.

$$

a_{n}=\int_{1}^{n} \frac{1}{x^{p}} d x, \quad p>1

$$

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Problem 85

The first term of a sequence is $x_{1}=1 .$ Each succeeding term is the sum of all those that come before it:

$$

x_{n+1}=x_{1}+x_{2}+\cdots+x_{n}

$$

Write out enough early terms of the sequence to deduce a general formula for $x_{n}$ that holds for $n \geq 2$ .

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Problem 86

A sequence of rational numbers is described as follows:

$$

\frac{1}{1}, \frac{3}{2}, \frac{7}{5}, \frac{17}{12}, \ldots, \frac{a}{b}, \frac{a+2 b}{a+b}, \ldots

$$

Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let $x_{n}$ and $y_{n}$ be, respectively, the numerator and the denominator of the $n$ th fraction $r_{n}=x_{n} / y_{n} .$

a. Verify that $x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1$ and, more generally, that if $a^{2}-2 b^{2}=-1$ or $+1,$ then

$$

(a+2 b)^{2}-2(a+b)^{2}=+1 \text { or }-1

$$

respectively.

b. The fractions $r_{n}=x_{n} / y_{n}$ approach a limit as $n$ increases. What is that limit? (Hint: Use part (a) to show that $r_{n}^{2}-2=\pm\left(1 / y_{n}\right)^{2}$ and that $y_{n}$ is not less than $n . )$

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Problem 87

Newton's method The following sequences come from the recursion formula for Newton's method,

$$

x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}

$$

Do the sequences converge? If so, to what value? In each case, begin by identifying the function $f$ that generates the sequence.

$$

\begin{array}{l}{\text { a. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{n}}} \\ {\text { b. } x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}} \\ {\text { c. } x_{0}=1, \quad x_{n+1}=x_{n}-1}\end{array}

$$

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Problem 88

a. Suppose that $f(x)$ is differentiable for all $x$ in $[0,1]$ and that $f(0)=0 .$ Define the sequence $\left\{a_{n}\right\}$ by the rule $a_{n}=$ $n f(1 / n) .$ Show that $\lim _{n \rightarrow \infty} a_{n}=f^{\prime}(0)$ Use the result in part (a) to find the limits of the following sequences $\left\{a_{n}\right\}$.

$\begin{array}{ll}{\text { b. } a_{n}=n \tan ^{-1} \frac{1}{n}} & {\text { c. } a_{n}=n\left(e^{1 / n}-1\right)} \\ {\text { d. } a_{n}=n \ln \left(1+\frac{2}{n}\right)}\end{array}$

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Problem 89

Pythagorean triples $A$ triple of positive integers $a, b,$ and $c$ is called a Pythagorean triple if $a^{2}+b^{2}=c^{2} .$ Let $a$ be an odd positive integer and let

$$

b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left[\frac{a^{2}}{2}\right]

$$

be, respectively, the integer floor and ceiling for $a^{2} / 2$.

Graph cannot copy

a. Show that $a^{2}+b^{2}=c^{2}$ . (Hint: Let $a=2 n+1$ and express $b$ and $c$ in terms of $n . )$

b. By direct calculation, or by appealing to the figure here, find

$$

\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil}

$$

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Problem 90

The $n$ th root of $n !$

a. Show that $\lim _{n \rightarrow \infty}(2 n \pi)^{1 /(2 n)}=1$ and hence, using Stirling's approximation (Chapter $8,$ Additional Exercise 50$a )$ , that

$$

\sqrt[n]{n !} \approx \frac{n}{e} \quad \text { for large values of } n

$$

b. Test the approximation in part (a) for $n=40,50,60, \ldots,$ as far as your calculator will allow.

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Problem 91

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive con-

$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0$$

if $c$ is any positive constant.

b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant. (Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to ensure that $\left|1 / n^{c}-0\right|<\epsilon$ if $n>N ? )$

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Problem 92

The zipper theorem Prove the "zipper theorem" for sequences: If $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ both converge to $L,$ then the sequence

$$

a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{n}, b_{n}, \ldots

$$

converges to $L$.

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Problem 97

In Exercises $97-100,$ determine if the sequence is nondecreasing and if it is bounded from above.

$$

a_{n}=\frac{3 n+1}{n+1}

$$

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Problem 98

In Exercises $97-100,$ determine if the sequence is nondecreasing and if it is bounded from above.

$$

a_{n}=\frac{(2 n+3) !}{(n+1) !}

$$

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Problem 99

In Exercises $97-100,$ determine if the sequence is nondecreasing and if it is bounded from above.

$$

a_{n}=\frac{2^{n} 3^{n}}{n !}

$$

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Problem 100

In Exercises $97-100,$ determine if the sequence is nondecreasing and if it is bounded from above.

$$

a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}}

$$

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Problem 101

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

$$

a_{n}=1-\frac{1}{n}

$$

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Problem 102

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

$$

a_{n}=n-\frac{1}{n}

$$

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Problem 103

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

$$

a_{n}=\frac{2^{n}-1}{2^{n}}

$$

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Problem 104

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

$$

a_{n}=\frac{2^{n}-1}{3^{n}}

$$

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Problem 105

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

$$

a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)

$$

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Problem 106

Which of the sequences in Exercises $101-106$ converge, and which diverge? Give reasons for your answers.

The first term of a sequence is $x_{1}=\cos (1) .$ The next terms are $x_{2}=x_{1}$ or $\cos (2),$ whichever is larger; and $x_{3}=x_{2}$ or $\cos (3),$ whichever is larger (farther to the right). In general,

$$

x_{n+1}=\max \left\{x_{n}, \cos (n+1)\right\}

$$

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Problem 107

Nonincreasing sequences A sequence of numbers $\left\{a_{n}\right\}$ in which $a_{n} \geq a_{n+1}$ for every $n$ is called a nonincreasing sequence. A sequence $\left\{a_{n}\right\}$ is bounded from below if there is a number $M$ with $M \leq a_{n}$ for every $n .$ Such a number $M$ is called a lower bound for the sequence. Deduce from Theorem 6 that a nonincreasing sequence that is bounded from below converges and that a nonincreasing sequence that is not bounded from below diverges.

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Problem 108

(Continuation of Exercise 107 ). Using the conclusion of Exercise 107 , determine which of the sequences in Exercises $108-112$ converge and which diverge.

$$

a_{n}=\frac{n+1}{n}

$$

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Problem 109

(Continuation of Exercise 107 ). Using the conclusion of Exercise 107 , determine which of the sequences in Exercises $108-112$ converge and which diverge.

$$

a_{n}=\frac{1+\sqrt{2 n}}{\sqrt{n}}

$$

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Problem 110

(Continuation of Exercise 107 ). Using the conclusion of Exercise 107 , determine which of the sequences in Exercises $108-112$ converge and which diverge.

$$

a_{n}=\frac{1-4^{n}}{2^{n}}

$$

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Problem 111

(Continuation of Exercise 107 ). Using the conclusion of Exercise 107 , determine which of the sequences in Exercises $108-112$ converge and which diverge.

$$

a_{n}=\frac{4^{n+1}+3^{n}}{4^{n}}

$$

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Problem 112

(Continuation of Exercise 107 ). Using the conclusion of Exercise 107 , determine which of the sequences in Exercises $108-112$ converge and which diverge.

$$

a_{1}=1, \quad a_{n+1}=2 a_{n}-3

$$

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Problem 113

The sequence $\{n /(n+1)\}$ has a least upper bound of 1 Show that if $M$ is a number less than $1,$ then the terms of $\{n /(n+1)\}$ eventually exceed $M .$ That is, if $M<1$ there is an integer $N$ such that $n /(n+1)>M$ whenever $n>N .$ since $n /(n+1)<1$ for every $n,$ this proves that 1 is a least upper bound for $\{n /(n+1)\}$.

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Problem 114

Uniqueness of least upper bounds Show that if $M_{1}$ and $M_{2}$ are least upper bounds for the sequence $\left\{a_{n}\right\},$ then $M_{1}=M_{2} .$ That is, a sequence cannot have two different least upper bounds.

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Problem 115

Is it true that a sequence $\left\{a_{n}\right\}$ of positive numbers must converge if it is bounded from above? Give reasons for your answer.

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Problem 116

Prove that if $\left\{a_{n}\right\}$ is a convergent sequence, then to every positive number $\epsilon$ there corresponds an integer $N$ such that for all $m$ and $n$ ,

$$

m>N \text { and } n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon

$$

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Problem 117

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if $L_{1}$ and $L_{2}$ are numbers such that $a_{n} \rightarrow L_{1}$ and $a_{n} \rightarrow L_{2},$ then $L_{1}=L_{2} .$

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Problem 118

Limits and subsequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a subsequence of the second. Prove that if two sub-sequences of a sequence $\left\{a_{n}\right\}$ have different limits $L_{1} \neq L_{2}$ then $\left\{a_{n}\right\}$ diverges.

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Problem 119

For a sequence $\left\{a_{n}\right\}$ the terms of even index are denoted by $a_{2 k}$ and the terms of odd index by $a_{2 k+1} .$ Prove that if $a_{2 k} \rightarrow L$ and $a_{2 k+1} \rightarrow L,$ then $a_{n} \rightarrow L .$

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Problem 120

Prove that a sequence $\left\{a_{n}\right\}$ converges to 0 if and only if the sequence of absolute values $\left\{\left|a_{n}\right|\right\}$ converges to 0 .

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Problem 121

In Exercises $121-124$ , experiment with a calculator to find a value of $N$ that will make the inequality hold for all $n>N$ . Assuming that the inequality is the one from the formal definition of the limit of a sequence, what sequence is being considered in each case and what is its limit?

$$

|\sqrt[n]{0.5}-1|<10^{-3}

$$

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Problem 122

In Exercises $121-124$ , experiment with a calculator to find a value of $N$ that will make the inequality hold for all $n>N$ . Assuming that the inequality is the one from the formal definition of the limit of a sequence, what sequence is being considered in each case and what is its limit?

$$

|\sqrt[n]{n}-1|<10^{-3}

$$

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Problem 123

In Exercises $121-124$ , experiment with a calculator to find a value of $N$ that will make the inequality hold for all $n>N$ . Assuming that the inequality is the one from the formal definition of the limit of a sequence, what sequence is being considered in each case and what is its limit?

$$

(0.9)^{n} < 10^{-3}

$$

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Problem 124

In Exercises $121-124$ , experiment with a calculator to find a value of $N$ that will make the inequality hold for all $n>N$ . Assuming that the inequality is the one from the formal definition of the limit of a sequence, what sequence is being considered in each case and what is its limit?

$$

2^{n} / n !<10^{-7}

$$

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Problem 125

Sequences generated by Newton's method Newton's method, applied to a differentiable function $f(x),$ begins with a starting value $x_{0}$ and constructs from it a sequence of numbers $\left\{x_{n}\right\}$ that under favorable circumstances converges to a zero of $f .$ The recursion formula for the sequence is

$$

x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)}

$$

a. Show that the recursion formula for $f(x)=x^{2}-a, a>0$ can be written as $x_{n+1}=\left(x_{n}+a / x_{n}\right) / 2$ b. Starting with $x_{0}=1$ and $a=3$ , calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

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Problem 126

(Continuation of Exercise $125 .$ ) Repeat part (b) of Exercise 125 with $a=2$ in place of $a=3$ .

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Problem 127

A recursive definition of $\pi / 2$ If you start with $x_{1}=1$ and define the subsequent terms of $\left\{x_{n}\right\}$ by the rule $x_{n}=x_{n-1}+\cos x_{n-1},$ you generate a sequence that converges rapidly to $\pi / 2 .$ a. Try it. b. Use the accompanying figure to explain why the convergence is so rapid.

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Problem 128

According to a front-page article in the December $15,1992,$ issue of the Wall Street Journal, Ford Motor Company used about 7$\frac{1}{4}$ hours of labor to produce stampings for the average vehicle, down from an estimated 15 hours in $1980 .$ The Japanese needed only about 3$\frac{1}{2}$ hours. Ford's improvement since 1980 represents an average decrease of 6$\%$ per year. If that rate continues, then $n$ years from 1992 Ford will use about

$$

S_{n}=7.25(0.94)^{n}

$$

hours of labor to produce stampings for the average vehicle. Assuming that the Japanese continue to spend 3$\frac{1}{2}$ hours per vehicle, how many more years will it take Ford to catch up? Find out two ways:

a. Find the first term of the sequence $\left\{S_{n}\right\}$ that is less than or equal to $3.5 .$

b. Graph $f(x)=7.25(0.94)^{x}$ and use Trace to find where the graph crosses the line $y=3.5 .$

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Problem 129

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\sqrt[n]{n}

$$

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Problem 130

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\left(1+\frac{0.5}{n}\right)^{n}

$$

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Problem 131

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{5^{n}}

$$

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Problem 132

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n}

$$

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Problem 133

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\sin n

$$

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Problem 134

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=n \sin \frac{1}{n}

$$

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Problem 135

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\frac{\sin n}{n}

$$

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Problem 136

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\frac{\ln n}{n}

$$

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Problem 137

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=(0.9999)^{n}

$$

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Problem 138

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=123456^{1 / n}

$$

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Problem 139

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\frac{8^{n}}{n !}

$$

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Problem 140

Use a CAS to perform the following steps for the sequences in Exercises $129-140 .$

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit $L ?$

b. If the sequence converges, find an integer $N$ such that $\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$

a_{n}=\frac{n^{41}}{19^{n}}

$$

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Problem 141

Compound interest, deposits, and withdrawals If you invest an amount of money $A_{0}$ at a fixed annual interest rate $r$ compounded $m$ times per year, and if the constant amount $b$ is added to the account at the end of each compounding period (or taken from the account if $b<0 ),$ then the amount you have after $n+1$ compounding periods is

$$

A_{n+1}=\left(1+\frac{r}{m}\right) A_{n}+b

$$

a. If $A_{0}=1000, r=0.02015, m=12,$ and $b=50$ , calculate and plot the first 100 points $\left(n, A_{n}\right) .$ How much money is in your account at the end of 5 years? Does $\left\{A_{n}\right\}$ converge? Is $\left\{A_{n}\right\}$ bounded?

b. Repeat part (a) with $A_{0}=5000, r=0.0589, m=12,$ and $b=-50 .$

c. If you invest 5000 dollars in a certificate of deposit (CD) that pays 4.5$\%$ annually, compounded quarterly, and you make no further investments in the CD, approximately how many years will it take before you have $20,000$ dollars? What if the CD earns 6.25$\% ?$

d. It can be shown that for any $k \geq 0$ , the sequence defined recursively by Equation $(1)$ satisfies the relation

$$

A_{k}=\left(1+\frac{r}{m}\right)^{k}\left(A_{0}+\frac{m b}{r}\right)-\frac{m b}{r}

$$

For the values of the constants $A_{0}, r, m,$ and $b$ given in part (a), validate this assertion by comparing the values of the first 50 terms of both sequences. Then show by direct substitution that the terms in Equation $(2)$ satisfy the recursion formula in Equation ( 1$)$ .

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Problem 142

Logistic difference equation The recursive relation

$$

a_{n+1}=r a_{n}\left(1-a_{n}\right)

$$

is called the logistic difference equation, and when the initial value $a_{0}$ is given the equation defines the logistic sequence $\left\{a_{n}\right\} .$ Throughout this exercise we choose $a_{0}$ in the interval $0<a_{0}<1,$ say $a_{0}=0.3$

a. Choose $r=3 / 4 .$ Calculate and plot the points $\left(n, a_{n}\right)$ for the first 100 terms in the sequence. Does it appear to converge? What do you guess is the limit? Does the limit seem to depend on your choice of $a_{0} ?$

b. Choose several values of $r$ in the interval $1< r <3$ and repeat the procedures in part (a). Be sure to choose some points near the endpoints of the interval. Describe the behavior of the sequences you observe in your plots.

c. Now examine the behavior of the sequence for values of $r$ near the endpoints of the interval $3< r <3.45 .$ The transition value $r=3$ is called a bifurcation value and the new behavior of the sequence in the interval is called an attracting 2 -cycle. Explain why this reasonably describes the behavior.

d. Next explore the behavior for $r$ values near the endpoints of each of the intervals $3.45< r <3.54$ and $3.54< r <3.55 .$ Plot the first 200 terms of the sequences. Describe in your own words the behavior observed in your plots for each interval. Among how many values does the sequence appear to oscillate for each interval? The values $r=3.45$ and $r=3.54$ (rounded to two decimal places) are also called bifurcation values because the behavior of the sequence changes as $r$ crosses over those values.

e. The situation gets even more interesting. There is actually an increasing sequence of bifurcation values $3<3.45<3.54$ $<\cdots<c_{n}<c_{n+1} \cdots$ such that for $c_{n}< r <c_{n+1}$ the logistic sequence $\left\{a_{n}\right\}$ eventually oscillates steadily among $2^{n}$ values, called an attracting $2^{n}$ -cycle. Moreover, the bifurcation sequence $\left\{c_{n}\right\}$ is bounded above by 3.57 (so it converges). If you choose a value of $r<3.57$ you will observe a $2^{n}$ -cycle of some sort. Choose $r=3.5695$ and plot 300 points.

f. Let us see what happens when $r>3.57$ . Choose $r=3.65$ and calculate and plot the first 300 terms of $\left\{a_{n}\right\} .$ Observe how the terms wander around in an unpredictable, chaotic fashion. You cannot predict the value of $a_{n+1}$ from previous values of the sequence.

g. For $r=3.65$ choose two starting values of $a_{0}$ that are close together, say, $a_{0}=0.3$ and $a_{0}=0.301 .$ Calculate and plot the first 300 values of the sequences determined by each starting value. Compare the behaviors observed in your plots. How far out do you go before the corresponding terms of your two sequences appear to depart from each other? Repeat the exploration for $r=3.75 .$ Can you see how the plots look different depending on your choice of $a_{0} ?$ We say that the logistic sequence is sensitive to the initial condition a_{0} .

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