Problem 1

Finding Terms of a Sequence

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

Finding Terms of a Sequence

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

Finding Terms of a Sequence

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

Finding Terms of a Sequence

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

Finding Terms of a Sequence

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

Finding Terms of a Sequence

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.

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

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Problem 13

Finding a Sequence's Formula In Exercises $13-26,$ 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 13-26, find a formula for the nth term of the sequence.

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

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Problem 15

In Exercises 13-26, find a formula for the nth term of the sequence.

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

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Problem 16

In Exercises 13-26, find a formula for the nth 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 $13-26,$ find a formula for the $n$ th term of the sequence.

$$\frac{1}{9}, \frac{2}{12}, \frac{2^{2}}{15}, \frac{2^{3}}{18}, \frac{2^{4}}{21}, \dots$$

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Problem 18

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

$$-\frac{3}{2},-\frac{1}{6}, \frac{1}{12}, \frac{3}{20}, \frac{5}{30}, \dots$$

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Problem 19

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

$$0,3,8,15,24, \ldots$$

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Problem 20

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

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

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Problem 21

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

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

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Problem 22

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

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

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Problem 22

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

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Problem 23

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

$$\frac{5}{1}, \frac{8}{2}, \frac{11}{6}, \frac{14}{24}, \frac{17}{120}, \ldots$$

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Problem 24

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

$$\frac{1}{25}, \frac{8}{125}, \frac{27}{625}, \frac{64}{3125}, \frac{125}{15,625}$$

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Problem 25

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

$$1,0,1,0,1, \ldots$$

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Problem 26

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.

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

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Problem 27

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 28

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which

diverge? Find the limit of each convergent sequence.

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

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Problem 29

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 30

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 31

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 32

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 33

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 34

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 35

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 36

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 37

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 38

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 39

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 40

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 41

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 42

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 43

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 44

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 45

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 46

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 47

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 48

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 49

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 50

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 51

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 52

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 53

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 54

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 55

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 56

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 57

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 58

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 59

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 60

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 61

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 62

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 63

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 64

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 65

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 66

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 67

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 68

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 69

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 70

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 71

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 72

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 74

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 75

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

$$a_{n}=\tanh n$$

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Problem 76

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 77

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 78

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 79

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 80

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

$$a_{n}=\left(3^{n}+5^{n}\right)^{1 / n}$$

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Problem 81

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 82

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 83

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 84

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 85

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 86

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 87

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and which diverge? Find the limit of each convergent sequence.

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

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Problem 88

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 89

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 90

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ 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 91

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}}$$

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Problem 92

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2}$$

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Problem 93

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=-4, \quad a_{n+1}=\sqrt{8+2 a_{n}}$$

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Problem 94

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=0, \quad a_{n+1}=\sqrt{8+2 a_{n}}$$

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Problem 95

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=5, \quad a_{n+1}=\sqrt{5} a_{n}$$

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Problem 96

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$a_{1}=3, \quad a_{n+1}=12-\sqrt{a_{n}}$$

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Problem 97

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots$$

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Problem 98

In Exercises $91-98,$ assume that each sequence converges and find its limit.

$$\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}$$

$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}$$

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Problem 99

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

$$x_{n+1}=x_{1}+x_{2}+\dots+x_{n}$$

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

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Problem 100

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 $m$ th fraction $r_{n}=x_{n} / y_{n}$

\begin{equation}\begin{array}{l}{\text { a. Verify that } x_{1}^{2}-2 y_{1}^{2}=-1, x_{2}^{2}-2 y_{2}^{2}=+1 \text { and, more }} \\ {\text { generally, that if } a^{2}-2 b^{2}=-1 \text { or }+1, \text { then }}\end{array}\end{equation}

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

respectively.

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

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Problem 101

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.

a. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{x_{n}^{2}-2}{2 x_{n}}=\frac{x_{n}}{2}+\frac{1}{x_{\mathrm{m}}}$

b. $x_{0}=1, \quad x_{n+1}=x_{n}-\frac{\tan x_{n}-1}{\sec ^{2} x_{n}}$

c. $x_{0}=1, \quad x_{n+1}=x_{2}-1$

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Problem 102

\begin{equation}\begin{array}{l}{\text { a. Suppose that } f(x) \text { is differentiable for all } x \text { in }[0,1] \text { and that }} \\ {f(0)=0 . \text { Define sequence }\left\{a_{n}\right\} \text { by the rule } a_{n}=n f(1 / n) \text { . }} \\ {\text { Show that lim }_{n \rightarrow \infty} a_{n}=f^{\prime}(0) . \text { Use the result in part (a) to }} \\ {\text { find the limits of the following sequences }\left\{a_{n}\right\}}\end{array} \\ {\text { b. } a_{n}=n \tan ^{-1} \frac{1}{n} \quad \text { c. } a_{n}=n\left(e^{1 / n}-1\right)} \\ {\text { d. } a_{n}=n \ln \left(1+\frac{2}{n}\right)}\end{equation}

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Problem 103

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

$b=\left\lfloor\frac{a^{2}}{2}\right\rfloor$ and $ c=\left\lceil\frac{a^{2}}{2}\right\rceil$

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

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 accompanying figure, find

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

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Problem 104

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 32 $\mathrm{a}$ , that

$\sqrt[n]{n !} \approx \frac{n}{a}$ for large values of $n$

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

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Problem 105

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

stant, show that

$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0$$ 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 106

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

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

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Problem 111

In Exercises $111-114,$ determine if the sequence is monotonic and if it is bounded.

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

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Problem 112

In Exercises $111-114,$ determine if the sequence is monotonic and if it is bounded.

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

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Problem 113

In Exercises $111-114,$ determine if the sequence is monotonic and if it is bounded.

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

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Problem 114

In Exercises $111-114,$ determine if the sequence is monotonic and if it is bounded.

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

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Problem 115

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 116

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 117

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 118

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 119

Which of the sequences in Exercises $115-124$ 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 120

Which of the sequences in Exercises $115-124$ 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 121

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 122

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 123

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 124

Which of the sequences in Exercises $115-124$ converge, and which diverge? Give reasons for your answers.

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

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Problem 125

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 aninteger $N$ 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 126

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 128

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 $n,$

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

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Problem 129

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 $a_{n} \rightarrow L_{2, \text { then }} L_{1}=L_{2}$ .

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Problem 130

Limits and sub sequences If the terms of one sequence appear

in another sequence in their given order, we call the first sequence a sub sequence 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 $\left\{a_{y}\right\}$ diverges.

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Problem 131

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 132

Prove that a sequence $\left\{a_{a}\right\}$ 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 133

Sequence. generated by N_on'. method Newton's method, applied to a different table function(x), begins with a starting value of and constructs from it a sequence of numbers {x,} 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 an proximated? Explain

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Problem 134

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 135

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 136

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

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

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Problem 137

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 138

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 139

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 140

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 141

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 142

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 143

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 144

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 145

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ 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 146

Use a CAS to perform the following steps for the sequences in Exerciscs $135-146 .$

a. Calculate and then plot the first 25 terms of the sequence Does the sequence appear to be bounded from above or be low? 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 $\left|a_{\pi}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do you have to get for the terms to lie within 0.0001 of $L ?$

$$\alpha_{n}=\frac{n^{41}}{19^{n}}$$

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