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Calculus for AP

Jon Rogawski & Ray Cannon

Chapter 10

INFINITE SERIES

Educators

FS

Problem 1

Match each sequence with its general term:

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Problem 2

Let $a_{n}=\frac{1}{2 n-1}$ for $n=1,2,3, \ldots$ Write out the first three terms of the following
sequences.
$$
\begin{array}{ll}{\text { (a) } b_{n}=a_{n+1}} & {\text { (b) } c_{n}=a_{n+3}} \\ {\text { (c) } d_{n}=a_{n}^{2}} & {\text { (d) } e_{n}=2 a_{n}-a_{n+1}}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 3

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
c_{n}=\frac{3^{n}}{n !}
$$

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Problem 4

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
b_{n}=\frac{(2 n-1) !}{n !}
$$

FS
Fuzail S.
Numerade Educator

Problem 5

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
a_{1}=2, \quad a_{n+1}=2 a_{n}^{2}-3
$$

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Problem 6

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
b_{1}=1, \quad b_{n}=b_{n-1}+\frac{1}{b_{n-1}}
$$

FS
Fuzail S.
Numerade Educator

Problem 7

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
b_{n}=5+\cos \pi n
$$

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Problem 8

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
c_{n}=(-1)^{2 n+1}
$$

FS
Fuzail S.
Numerade Educator

Problem 9

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
c_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}
$$

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Problem 10

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
a_{n}=n+(n+1)+(n+2)+\cdots+(2 n)
$$

FS
Fuzail S.
Numerade Educator

Problem 11

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
b_{1}=2, \quad b_{2}=3, \quad b_{n}=2 b_{n-1}+b_{n-2}
$$

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Problem 12

In Exercises $3-12,$ calculate the first four terms of the sequence, starting with $n=1$
$$
c_{n}=n-\text { place decimal approximation to } e
$$

FS
Fuzail S.
Numerade Educator

Problem 13

Find a formula for the $n$ th term of each sequence.
$$
\text { (a) } \frac{1}{1}, \frac{-1}{8}, \frac{1}{27}, \ldots \quad \text { (b) } \frac{2}{6}, \frac{3}{7}, \frac{4}{8}, \ldots
$$

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Problem 14

Suppose that $\lim _{n \rightarrow \infty} a_{n}=4$ and $\lim _{n \rightarrow \infty} b_{n}=7 .$ Determine:
$$
\begin{array}{ll}{\text { (a) } \lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)} & {\text { (b) } \lim _{n \rightarrow \infty} a_{n}^{3}} \\ {(\mathrm{c})} {\lim _{n \rightarrow \infty} \cos \left(\pi b_{n}\right)} & {\text { (d) } \lim _{n \rightarrow \infty}\left(a_{n}^{2}-2 a_{n} b_{n}\right)}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 15

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=12
$$

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Problem 16

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=20-\frac{4}{n^{2}}
$$

FS
Fuzail S.
Numerade Educator

Problem 17

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
b_{n}=\frac{5 n-1}{12 n+9}
$$

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Problem 18

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=\frac{4+n-3 n^{2}}{4 n^{2}+1}
$$

FS
Fuzail S.
Numerade Educator

Problem 19

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
c_{n}=-2^{-n}
$$

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Problem 20

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
z_{n}=\left(\frac{1}{3}\right)^{n}
$$

FS
Fuzail S.
Numerade Educator

Problem 21

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
c_{n}=9^{n}
$$

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Problem 22

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
z_{n}=10^{-1 / n}
$$

FS
Fuzail S.
Numerade Educator

Problem 23

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=\frac{n}{\sqrt{n^{2}+1}}
$$

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Problem 24

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=\frac{n}{\sqrt{n^{3}+1}}
$$

FS
Fuzail S.
Numerade Educator

Problem 25

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
a_{n}=\ln \left(\frac{12 n+2}{-9+4 n}\right)
$$

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Problem 26

In Exercises $15-26,$ use Theorem I to determine the limit of the sequence or state that the sequence diverges.
$$
r_{n}=\ln n-\ln \left(n^{2}+1\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 27

In Exercises $27-30,$ use Theorem 4 to determine the limit of the sequence.
$$
a_{n}=\sqrt{4+\frac{1}{n}}
$$

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Problem 28

In Exercises $27-30,$ use Theorem 4 to determine the limit of the sequence.
$$
a_{n}=e^{4 n /(3 n+9)}
$$

FS
Fuzail S.
Numerade Educator

Problem 29

In Exercises $27-30,$ use Theorem 4 to determine the limit of the sequence.
$$
a_{n}=\cos ^{-1}\left(\frac{n^{3}}{2 n^{3}+1}\right)
$$

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Problem 30

In Exercises $27-30,$ use Theorem 4 to determine the limit of the sequence.
$$
a_{n}=\tan ^{-1}\left(e^{-n}\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 31

Let $a_{n}=\frac{n}{n+1} .$ Find a number $M$ such that:
$$
\begin{array}{l}{\text { (a) Find a value of } M \text { such that }\left|b_{n}\right| \leq 10^{-5} \text { for } n \geq M} \\ {\text { (b) Use the limit definition to prove that } \lim _{n \rightarrow \infty} b_{n}=0}\end{array}
$$

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Problem 32

Let $$b_{n}=\left(\frac{1}{3}\right)^{n}$$
$$
\begin{array}{l}{\text { (a) Find a value of } M \text { such that }\left|b_{n}\right| \leq 10^{-5} \text { for } n \geq M} \\ {\text { (b) Use the limit definition to prove that } \lim _{n \rightarrow \infty} b_{n}=0}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 33

Use the limit definition to prove that $\lim _{n \rightarrow \infty} n^{-2}=0$

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Problem 34

Use the limit definition to prove that $\lim _{n \rightarrow \infty} \frac{n}{n+n^{-1}}=1$

FS
Fuzail S.
Numerade Educator

Problem 35

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=10+\left(-\frac{1}{9}\right)^{n}
$$

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Problem 36

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
d_{n}=\sqrt{n+3}-\sqrt{n}
$$

FS
Fuzail S.
Numerade Educator

Problem 37

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
c_{n}=1.01^{n}
$$

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Problem 38

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=e^{1-n^{2}}
$$

FS
Fuzail S.
Numerade Educator

Problem 39

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=2^{1 / n}
$$

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Problem 40

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=n^{1 / n}
$$

FS
Fuzail S.
Numerade Educator

Problem 41

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
c_{n}=\frac{9^{n}}{n !}
$$

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Problem 42

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{8^{2 n}}{n !}
$$

FS
Fuzail S.
Numerade Educator

Problem 43

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{3 n^{2}+n+2}{2 n^{2}-3}
$$

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Problem 44

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{\sqrt{n}}{\sqrt{n}+4}
$$

FS
Fuzail S.
Numerade Educator

Problem 45

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{\cos n}{n}
$$

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Problem 46

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
c_{n}=\frac{(-1)^{n}}{\sqrt{n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 47

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
d_{n}=\ln 5^{n}-\ln n !
$$

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Problem 48

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
d_{n}=\ln \left(n^{2}+4\right)-\ln \left(n^{2}-1\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 49

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\left(2+\frac{4}{n^{2}}\right)^{1 / 3}
$$

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Problem 50

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=\tan ^{-1}\left(1-\frac{2}{n}\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 51

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
c_{n}=\ln \left(\frac{2 n+1}{3 n+4}\right)
$$

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Problem 52

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
c_{n}=\frac{n}{n+n^{1 / n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 53

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
y_{n}=\frac{e^{n}}{2^{n}}
$$

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Problem 54

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{n}{2^{n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 55

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
y_{n}=\frac{e^{n}+(-3)^{n}}{5^{n}}
$$

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Problem 56

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=\frac{(-1)^{n} n^{3}+2^{-n}}{3 n^{3}+4^{-n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 57

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=n \sin \frac{\pi}{n}
$$

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Problem 58

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=\frac{n !}{\pi^{n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 59

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
b_{n}=\frac{3-4^{n}}{2+7 \cdot 4^{n}}
$$

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Problem 60

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\frac{3-4^{n}}{2+7 \cdot 3^{n}}
$$

FS
Fuzail S.
Numerade Educator

Problem 61

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\left(1+\frac{1}{n}\right)^{n}
$$

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Problem 62

In Exercises $35-62,$ use appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
$$
a_{n}=\left(1+\frac{1}{n^{2}}\right)^{n}
$$

FS
Fuzail S.
Numerade Educator

Problem 63

In Exercises $63-66,$ find the limit of the sequence using L'Hopitals
Rule.
$$
a_{n}=\frac{(\ln n)^{2}}{n}
$$

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Problem 64

In Exercises $63-66,$ find the limit of the sequence using L'Hopitals
Rule.
$$
b_{n}=\sqrt{n} \ln \left(1+\frac{1}{n}\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 65

In Exercises $63-66,$ find the limit of the sequence using L'Hopitals
Rule.
$$
c_{n}=n\left(\sqrt{n^{2}+1}-n\right)
$$

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Problem 66

In Exercises $63-66,$ find the limit of the sequence using L'Hopitals
Rule.
$$
d_{n}=n^{2}\left(\sqrt[3]{n^{3}+1}-n\right)
$$

FS
Fuzail S.
Numerade Educator

Problem 67

In Exercises $67-70,$ use the Squeeze Theorem to evaluate $\lim _{n \rightarrow \infty} a_{n} b y$
verifying the given inequality.
$$
a_{n}=\frac{1}{\sqrt{n^{4}+n^{8}}}, \quad \frac{1}{\sqrt{2} n^{4}} \leq a_{n} \leq \frac{1}{\sqrt{2} n^{2}}
$$

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Problem 68

In Exercises $67-70,$ use the Squeeze Theorem to evaluate $\lim _{n \rightarrow \infty} a_{n} b y$
verifying the given inequality.
$$
\begin{array}{l}{c_{n}=\frac{1}{\sqrt{n^{2}+1}}+\frac{1}{\sqrt{n^{2}+2}}+\dots+\frac{1}{\sqrt{n^{2}+n}}} \\ {\frac{n}{\sqrt{n^{2}+n}} \leq c_{n} \leq \frac{n}{\sqrt{n^{2}+1}}}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 69

In Exercises $67-70,$ use the Squeeze Theorem to evaluate $\lim _{n \rightarrow \infty} a_{n} b y$
verifying the given inequality.
$$
a_{n}=\left(2^{n}+3^{n}\right)^{1 / n}, \quad 3 \leq a_{n} \leq\left(2 \cdot 3^{n}\right)^{1 / n}=2^{1 / n} \cdot 3
$$

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Problem 70

In Exercises $67-70,$ use the Squeeze Theorem to evaluate $\lim _{n \rightarrow \infty} a_{n} b y$
verifying the given inequality.
$$
a_{n}=\left(n+10^{n}\right)^{1 / n}, \quad 10 \leq a_{n} \leq\left(2 \cdot 10^{n}\right)^{1 / n}
$$

FS
Fuzail S.
Numerade Educator

Problem 71

Which of the following statements is equivalent to the assertion $\lim _{n \rightarrow \infty} a_{n}=L ?$ Explain.
$$
\begin{array}{l}{\text { (a) For every } \epsilon>0, \text { the interval }(L-\epsilon, L+\epsilon) \text { contains at least one }} \\ {\text { element of the sequence }\left\{a_{n}\right\} .} \\ {\text { (b) For every } \epsilon>0, \text { the interval }(L-\epsilon, L+\epsilon) \text { contains all but at }} \\ {\text { most finitely many elements of the sequence }\left\{a_{n}\right\} .}\end{array}
$$

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Problem 72

Show that $a_{n}=\frac{1}{2 n+1}$ is decreasing.

FS
Fuzail S.
Numerade Educator

Problem 73

Show that $a_{n}=\frac{3 n^{2}}{n^{2}+2}$ is increasing. Find an upper bound.

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Problem 74

Show that $a_{n}=\sqrt[3]{n+1}-n$ is decreasing.

FS
Fuzail S.
Numerade Educator

Problem 75

Give an example of a divergent sequence $\left\{a_{n}\right\}$ such that $\lim _{n \rightarrow \infty}\left|a_{n}\right|$ converges.

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Problem 76

Give an example of divergent sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ such that
$\left\{a_{n}+b_{n}\right\}$ converges.

FS
Fuzail S.
Numerade Educator

Problem 77

Using the limit definition, prove that if $\left\{a_{n}\right\}$ converges and $\left\{b_{n}\right\}$
diverges, then $\left\{a_{n}+b_{n}\right\}$ diverges.

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Problem 78

Use the limit definition to prove that if $\left\{a_{n}\right\}$ is a convergent sequence of integers with limit $L,$ then there exists a number $M$ such that $a_{n}=L$ for all $n \geq M .$

FS
Fuzail S.
Numerade Educator

Problem 79

Theorem 1 states that if $\lim _{x \rightarrow \infty} f(x)=L,$ then the sequence $a_{n}=$ $f(n)$ converges and $\lim _{n \rightarrow \infty} a_{n}=L .$ Show that the converse is false. In other words, find a function $f(x)$ such that $a_{n}=f(n)$ converges but $\lim _{x \rightarrow \infty} f(x)$ does not exist.

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Problem 80

Use the limit definition to prove that the limit does not change if a finite number of terms are added or removed from a convergent sequence.

FS
Fuzail S.
Numerade Educator

Problem 81

Let $b_{n}=a_{n+1} .$ Use the limit definition to prove that if $\left\{a_{n}\right\}$ converges, then $\left\{b_{n}\right\}$ also converges and $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}$

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Problem 82

Let $\left\{a_{n}\right\}$ be a sequence such that $\lim _{n \rightarrow \infty}\left|a_{n}\right|$ exists and is nonzero. Show that $\lim _{n \rightarrow \infty} a_{n}$ exists if and only if there exists an integer $M$ such that the sign of $a_{n}$ does not change for $n>M$

FS
Fuzail S.
Numerade Educator

Problem 83

Proceed as in Example 12 to show that the sequence $\sqrt{3}, \sqrt{3 \sqrt{3}}$ , $\sqrt{3 \sqrt{3} \sqrt{3},}, \ldots$ is increasing and bounded above by $M=3 .$ Then prove that the limit exists and find its value.

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Problem 84

Let $\left\{a_{n}\right\}$ be the sequence defined recursively by
$$
a_{0}=0, \quad a_{n+1}=\sqrt{2+a_{n}}
$$
Thus, $a_{1}=\sqrt{2}, \quad a_{2}=\sqrt{2+\sqrt{2}}, \quad a_{3}=\sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots$
$$
\begin{array}{l}{\text { (a) Show that if } a_{n}<2, \text { then } a_{n+1}<2 . \text { Conclude by induction that }} \\ {a_{n}<2 \text { for all } n .} \\ {\text { (b) Show that if } a_{n}<2, \text { then } a_{n} \leq a_{n+1} . \text { Conclude by induction that }} \\ {\left\{a_{n}\right\} \text { is increasing. }} \\ {\text { (c) Use (a) and (b) to conclude that } L=\lim _{n \rightarrow \infty} a_{n} \text { exists. Then compute }} \\ {L \text { by showing that } L=\sqrt{2+L}}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 85

Show that $\lim _{n \rightarrow \infty} \sqrt[n]{n !}=\infty .$ Hint: Verify that $n ! \geq(n / 2)^{n / 2}$ by observing that half of the factors of $n !$ are greater than or equal to $n / 2$ .

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Problem 86

Let $$b_{n}=\frac{\sqrt[n]{n !}}{n}$$
$$
\begin{array}{l}{\text { (a) Show that } \ln b_{n}=\frac{1}{n} \sum_{k=1}^{n} \ln \frac{k}{n} \text { . }} \\ {\text { (b) Show that } \ln b_{n} \text { converges to } \int_{0}^{1} \ln x d x, \text { and conclude that }} \\ {b_{n} \rightarrow e^{-1}}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 87

Given positive numbers $a_{1} < b_{1},$ define two sequences recursively by
$$
a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2}
$$
$$
\begin{array}{l}{\text { (a) Show that } a_{n} \leq b_{n} \text { for all } n \text { (Figure } 13 ) .} \\ {\text { (b) Show that }\left\{a_{n}\right\} \text { is increasing and }\left\{b_{n}\right\} \text { is decreasing. }} \\ {\text { (c) Show that } b_{n+1}-a_{n+1} \leq \frac{b_{n}-a_{n}}{2}}\\{\text { (d) Prove that both }\left\{a_{n}\right\} \text { and }\left\{b_{n}\right\} \text { converge and have the same limit. }} \\ {\text { This limit, denoted } \operatorname{AGM}\left(a_{1}, b_{1}\right), \text { is called the arithmetic-geometric }} \\ {\text { mean of } a_{1} \text { and } b_{1} \text { . }} \\ {\text { (e) Estimate } \mathrm{AGM}(1, \sqrt{2}) \text { to three decimal places. }}\end{array}
$$

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Problem 88

Let $$c_{n}=\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n}$$
$$
\begin{array}{l}{\text { (a) Calculate } c_{1}, c_{2}, c_{3}, c_{4} \text { . }} \\ {\text { (b) Use a comparison of rectangles with the area under } y=x^{-1} \text { over }} \\ {\text { the interval }[n, 2 n] \text { to prove that }}\\{\int_{n}^{2 n} \frac{d x}{x}+\frac{1}{2 n} \leq c_{n} \leq \int_{n}^{2 n} \frac{d x}{x}+\frac{1}{n}} \\ {\text { (c) Use the Squeeze Theorem to determine } \lim _{n \rightarrow \infty} c_{n}}\end{array}
$$

FS
Fuzail S.
Numerade Educator

Problem 89

Let $a_{n}=H_{n}-\ln n,$ where $H_{n}$ is the $n$ th harmonic number
$$
H_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}
$$
$$
\begin{array}{l}{\text { (a) Show that } a_{n} \geq 0 \text { for } n \geq 1 . \text { Hint: Show that } H_{n} \geq \int_{1}^{n+1} \frac{d x}{x} \text { . }} \\ {\text { (b) Show that }\left\{a_{n}\right\} \text { is decreasing by interpreting } a_{n}-a_{n+1} \text { as an area. }} \\ {\text { (c) Prove that } \lim _{n \rightarrow \infty} a_{n} \text { exists. }}\end{array}
$$
This limit, denoted $\gamma,$ is known as Euler's Constant. It appears in many
areas of mathematics, including analysis and number theory, and has
been calculated to more than 100 million decimal places, but it is still
not known whether $\gamma$ is an irrational number. The first 10 digits are
$\gamma \approx 0.5772156649$

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