# Calculus for AP

## 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.
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
not known whether $\gamma$ is an irrational number. The first 10 digits are
$\gamma \approx 0.5772156649$