Thomas Calculus

George B. Thomas, Jr.

Chapter 10

Infinite Sequences and Series - all with Video Answers

Educators

+ 7 more educators

Section 1

Sequences

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

Sanchit Jain

Sanchit Jain

Numerade Educator

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 $a_{4} .$
$$
a_{n}=\frac{1}{n !}
$$

Willis James

Numerade Educator

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 $a_{4} .$
$$
a_{n}=\frac{(-1)^{n+1}}{2 n-1}
$$

Bobby Barnes

University of North Texas

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 $a_{4} .$
$$
a_{n}=2+(-1)^{n}
$$

Sanchit Jain

Numerade Educator

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 $a_{4} .$
$$
a_{n}=\frac{2^{n}}{2^{n+1}}
$$

William Semus

Numerade Educator

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

Bobby Barnes

University of North Texas

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

Bobby Barnes

University of North Texas

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

Bowen Gang

Numerade Educator

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

Bobby Barnes

University of North Texas

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

Bowen Gang

Numerade Educator

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

Bowen Gang

Numerade Educator

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

Bobby Barnes

University of North Texas

Problem 13

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $1,-1,1,-1,1, \ldots$

William Semus

Numerade Educator

Problem 14

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $-1,1,-1,1,-1, \ldots$

William Semus

Numerade Educator

Problem 15

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $1,-4,9,-16,25, \dots$

Zachary Mitchell

Zachary Mitchell

Numerade Educator

Problem 16

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}, \dots$

Zachary Mitchell

Zachary Mitchell

Numerade Educator

Problem 17

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

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

Chris Trentman

Numerade Educator

Problem 18

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
$$
-\frac{3}{2},-\frac{1}{6}, \frac{1}{12}, \frac{3}{20}, \frac{5}{30}, \dots
$$

Chris Trentman

Numerade Educator

Problem 19

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $0,3,8,15,24, \dots$

Chris Trentman

Numerade Educator

Problem 20

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $-3,-2,-1,0,1, \ldots$

Chris Trentman

Numerade Educator

Problem 21

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $1,5,9,13,17, \dots$

Andrew Sum

Numerade Educator

Problem 22

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $2,6,10,14,18, \dots$

Andrew Sum

Numerade Educator

Problem 23

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
$$
\frac{5}{1}, \frac{8}{2}, \frac{11}{6}, \frac{14}{24}, \frac{17}{120}, \dots
$$

Chris Trentman

Numerade Educator

Problem 24

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
$$
\frac{1}{25}, \frac{8}{125}, \frac{27}{625}, \frac{64}{3125}, \frac{125}{15,625}, \ldots
$$

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

Problem 25

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $1,0,1,0,1, \ldots$

Chris Trentman

Numerade Educator

Problem 26

In Exercises $13-26,$ find a formula for the $n$ th term of the sequence.
The sequence $0,1,1,2,2,3,3,4, \ldots$

Andrew Sum

Numerade Educator

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

Willis James

Numerade Educator

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

Chris Trentman

Numerade Educator

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

AM

Amr Mansour

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Zhumagali Shomanov

Zhumagali Shomanov

Numerade Educator

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

Aman Gupta

Numerade Educator

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

AM

Amr Mansour

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

AM

Amr Mansour

Numerade Educator

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

AM

Amr Mansour

Numerade Educator

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

AM

Amr Mansour

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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.

Victor Salazar

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Aman Gupta

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

Problem 73

Which of the sequences $\left\{a_{n}\right\}$ in Exercises $27-90$ converge, and
which diverge? Find the limit of each convergent sequence.
$$
a_{n}=\frac{3^{n} \cdot 6^{n}}{2^{-n} \cdot n !}
$$

Aman Gupta

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Hoan Nguyen

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

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

Claire Rochford

Claire Rochford

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

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

Chris Trentman

Numerade Educator

Problem 91

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}}
$$

Chris Trentman

Numerade Educator

Problem 92

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=-1, \quad a_{n+1}=\frac{a_{n}+6}{a_{n}+2}
$$

Chris Trentman

Numerade Educator

Problem 93

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=-4, \quad a_{n+1}=\sqrt{8+2 a_{n}}
$$

Chris Trentman

Numerade Educator

Problem 94

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=0, \quad a_{n+1}=\sqrt{8+2 a_{n}}
$$

Chris Trentman

Numerade Educator

Problem 95

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=5, \quad a_{n+1}=\sqrt{5 a_{n}}
$$

Chris Trentman

Numerade Educator

Problem 96

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
a_{1}=3, \quad a_{n+1}=12-\sqrt{a_{n}}
$$

Chris Trentman

Numerade Educator

Problem 97

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
2,2+\frac{1}{2}, 2+\frac{1}{2+\frac{1}{2}}, 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}, \dots
$$

Chris Trentman

Numerade Educator

Problem 98

In Exercises $91-98$ , assume that each sequence converges and find its
limit.
$$
\begin{array}{l}{\sqrt{1}, \sqrt{1+\sqrt{1}}, \sqrt{1+\sqrt{1+\sqrt{1}}}} \\ {\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}, \ldots}\end{array}
$$

Jacob Fry

Numerade Educator

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 $n \geq 2$ .

Bobby Barnes

University of North Texas

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

Vishal Parmar

Numerade Educator

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

Bobby Barnes

University of North Texas

Problem 102

a. Suppose that $f(x)$ is differentiable for all $x$ in $[0,1]$ and that
$f(0)=0 .$ Define 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 $\left\{a_{n}\right\} .$
$$\begin{aligned} \text { b. } a_{n} &=n \tan ^{-1} \frac{1}{n} & \text { c. } a_{n}=n\left(e^{1 / n}-1\right) \\ \text { d. } a_{n} &=n \ln \left(1+\frac{2}{n}\right) \end{aligned}$$

Nick Johnson

Numerade Educator

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
$$
b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \text { and } c=\left\lceil\frac{a^{2}}{2}\right\rceil
$$
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
$$
\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{ | \frac{a^{2}}{2} \rceil}
$$

Yingtai Xiao

Numerade Educator

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 52 $\mathrm{a}$ , that
$$\sqrt[n]{n !}=\frac{n}{e} \quad$ for large values of $n$$
b. Test the approximation in part (a) for $n=40,50,60, \ldots,$ as
far as your calculator will allow.

Victor Salazar

Numerade Educator

Problem 105

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

Nick Johnson

Numerade Educator

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
$$a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}, \dots$$
converges to $L$

Sajay Krishnan Paruthiyil

Sajay Krishnan Paruthiyil

Numerade Educator

Problem 107

Prove that $\lim _{n \rightarrow \infty} \sqrt[n]{n}=1$

Vishal Parmar

Numerade Educator

Problem 108

$$
\lim _{n \rightarrow \infty} x^{1 / n}=1,(x>0)
$$

Hoan Nguyen

Numerade Educator

Problem 109

Prove Theorem 2

Nick Johnson

Numerade Educator

Problem 110

Prove Theorem 3

Victor Salazar

Numerade Educator

Problem 111

In Exercises $111-114,$ determine if the sequence is monotonic and if it
is bounded.
$$a_{n}=\frac{3 n+1}{n+1}$$

Chris Trentman

Numerade Educator

Problem 112

In Exercises $111-114,$ determine if the sequence is monotonic and if it
is bounded.
$$
a_{n}=\frac{(2 n+3) !}{(n+1) !}
$$

Chris Trentman

Numerade Educator

Problem 113

In Exercises $111-114,$ determine if the sequence is monotonic and if it
is bounded.
$$
a_{n}=\frac{2^{n} 3^{n}}{n !}
$$

Chris Trentman

Numerade Educator

Problem 114

In Exercises $111-114,$ determine if the sequence is monotonic and if it
is bounded.
$$
a_{n}=2-\frac{2}{n}-\frac{1}{2^{n}}
$$

Chris Trentman

Numerade Educator

Problem 115

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$a_{n}=1-\frac{1}{n}$$

Chris Trentman

Numerade Educator

Problem 116

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$
a_{n}=n-\frac{1}{n}
$$

Chris Trentman

Numerade Educator

Problem 117

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$
a_{n}=\frac{2^{n}-1}{2^{n}}
$$

Bobby Barnes

University of North Texas

Problem 118

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$
a_{n}=\frac{2^{n}-1}{3^{n}}
$$

Chris Trentman

Numerade Educator

Problem 119

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$
a_{n}=\left((-1)^{n}+1\right)\left(\frac{n+1}{n}\right)
$$

Chris Trentman

Numerade Educator

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

Victor Salazar

Numerade Educator

Problem 121

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$a_{n}=\frac{1+\sqrt{2 n}}{\sqrt{n}}$$

Chris Trentman

Numerade Educator

Problem 122

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$a_{n}=\frac{n+1}{n}$

Chris Trentman

Numerade Educator

Problem 123

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$a_{n}=\frac{4^{n+1}+3^{n}}{4^{n}}$$

Bobby Barnes

University of North Texas

Problem 124

Which of the sequences in Exercises $115-124$ converge, and which
diverge? Give reasons for your answers.
$$a_{1}=1, \quad a_{n+1}=2 a_{n}-3$$

Chris Trentman

Numerade Educator

Problem 125

In Exercises $125-126,$ use the definition of convergence to prove the
given limit.
$$\lim _{n \rightarrow \infty} \frac{\sin n}{n}=0$$

Hoan Nguyen

Numerade Educator

Problem 126

In Exercises $125-126,$ use the definition of convergence to prove the
given limit.
$$\lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)=1$$

Hoan Nguyen

Numerade Educator

Problem 127

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 $\{n /(n+1)\} .$

Bobby Barnes

University of North Texas

Problem 128

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.

Vishal Parmar

Numerade Educator

Problem 129

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.

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 130

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

Zachary Mitchell

Zachary Mitchell

Numerade Educator

Problem 131

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 $L_{1}=L_{2}$ .

Bobby Barnes

University of North Texas

Problem 132

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.

Nick Johnson

Numerade Educator

Problem 133

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 $a_{n} \rightarrow L$

Muhammad Saleem

Muhammad Saleem

Numerade Educator

Problem 134

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 $0 .$

Chris Trentman

Numerade Educator

Problem 135

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.

Bobby Barnes

University of North Texas

Problem 136

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.

Nick Johnson

Numerade Educator

Problem 137

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$a_{n}=\sqrt[n]{n}$$

Lucas Finney

Numerade Educator

Problem 138

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\left(1+\frac{0.5}{n}\right)^{n}
$$

Lucas Finney

Numerade Educator

Problem 139

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{1}=1, \quad a_{n+1}=a_{n}+\frac{1}{5^{n}}
$$

Patrick Delos Reyes

Patrick Delos Reyes

Numerade Educator

Problem 140

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n}
$$

Lucas Finney

Numerade Educator

Problem 141

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\sin n
$$

Lucas Finney

Numerade Educator

Problem 142

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=n \sin \frac{1}{n}
$$

Lucas Finney

Numerade Educator

Problem 143

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\frac{\sin n}{n}
$$

Lucas Finney

Numerade Educator

Problem 144

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\frac{\ln n}{n}
$$

Lucas Finney

Numerade Educator

Problem 145

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=(0.9999)^{n}
$$

Lucas Finney

Numerade Educator

Problem 146

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=(123456)^{1 / n}
$$

Lucas Finney

Numerade Educator

Problem 147

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\frac{8^{n}}{n !}
$$

Lucas Finney

Numerade Educator

Problem 148

Use a CAS to perform the following steps for the sequences in Exercises $137-148 .$
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
$\quad\left|a_{n}-L\right| \leq 0.01$ for $n \geq N .$ How far in the sequence do
you have to get for the terms to lie within 0.0001 of $L ?$
$$
a_{n}=\frac{n^{41}}{19^{n}}
$$

Lucas Finney

Numerade Educator