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Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 9

Infinite Series - all with Video Answers

Educators

Section 1

Sequences

00:39

Problem 1

Write the first five terms of the sequence.
$$a_{n}=3^{n}$$

Maninder Singh
Maninder Singh
Numerade Educator
01:01

Problem 2

Write the first five terms of the sequence.
$$a_{n}=\left(-\frac{2}{5}\right)^{n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 3

Write the first five terms of the sequence.
$$a_{n}=\sin \frac{n \pi}{2}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 4

Write the first five terms of the sequence.
$$a_{n}=\frac{3 n}{n+4}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 5

Write the first five terms of the sequence.
$$a_{n}=(-1)^{n+1}\left(\frac{2}{n}\right)$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 6

Write the first five terms of the sequence.
$$a_{n}=2+\frac{2}{n}-\frac{1}{n^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:37

Problem 7

Write the first five terms of the recursively defined sequence.
$$a_{1}=3, a_{k+1}=2\left(a_{k}-1\right)$$

AG
Ankit Gupta
Numerade Educator
01:02

Problem 8

Write the first five terms of the recursively defined sequence.
$$a_{1}=6, a_{k+1}=\frac{1}{3} a_{k}^{2}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:20

Problem 9

Match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$a_{n}=\frac{10}{n+1}$$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:20

Problem 10

Match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$a_{n}=\frac{10 n}{n+1}$$

Anurag Kumar
Anurag Kumar
Numerade Educator
04:11

Problem 11

Match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$a_{n}=(-1)^{n}$$

Andrija Isakov
Andrija Isakov
Numerade Educator
04:11

Problem 12

Match the sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] (GRAPH CAN'T COPY)
$$a_{n}=\frac{(-1)^{n}}{n}$$

Andrija Isakov
Andrija Isakov
Numerade Educator
01:01

Problem 13

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$2,5,8,11, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 14

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$8,13,18,23,28, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 15

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$5,10,20,40, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 16

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms.
$$6,-2, \frac{2}{3},-\frac{2}{9}, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 17

Simplify the ratio of factorials.
$$\frac{(n+1) !}{n !}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
00:34

Problem 18

Simplify the ratio of factorials.
$$\frac{n !}{(n+2) !}$$

Amy Jiang
Amy Jiang
Numerade Educator
01:09

Problem 19

Simplify the ratio of factorials.
$$\frac{(2 n-1) !}{(2 n+1) !}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 20

Simplify the ratio of factorials.
$$\frac{(2 n+2) !}{(2 n) !}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 21

Find the limit (if possible) of the sequence.
$$a_{n}=\frac{5 n^{2}}{n^{2}+2}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 22

Find the limit (if possible) of the sequence.
$$a_{n}=6+\frac{2}{n^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 23

Find the limit (if possible) of the sequence.
$$a_{n}=\frac{2 n}{\sqrt{n^{2}+1}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 24

Find the limit (if possible) of the sequence.
$$a_{n}=\cos \frac{2}{n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:24

Problem 25

Use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$a_{n}=\frac{4 n+1}{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:17

Problem 26

Use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$a_{n}=\frac{1}{n^{3 / 2}}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:35

Problem 27

Use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$a_{n}=\sin \frac{n \pi}{2}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
02:05

Problem 28

Use a graphing utility to graph the first 10 terms of the sequence. Use the graph to make an inference about the convergence or divergence of the sequence. Verify your inference analytically and, if the sequence converges, find its limit.
$$a_{n}=2-\frac{1}{4^{n}}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:01

Problem 29

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{5}{n+2}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 30

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=8+\frac{5}{n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:13

Problem 31

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 32

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{1+(-1)^{n}}{n^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:03

Problem 33

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{10 n^{2}+3 n+7}{2 n^{2}-6}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 34

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{\sqrt[3]{n}}{\sqrt[3]{n}+1}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 35

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{\ln \left(n^{3}\right)}{2 n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 36

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{5^{n}}{3^{n}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 37

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{(n+1) !}{n !}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 38

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{(n-2) !}{n !}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:04

Problem 39

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{n^{p}}{e^{n}}, p>0$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:05

Problem 40

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=n \sin \frac{1}{n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 41

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=2^{1 / n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 42

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=-3^{-n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 43

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{\sin n}{n}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 44

Determine the convergence or divergence of the sequence with the given $n$ th term. If the sequence converges, find its limit.
$$a_{n}=\frac{\cos \pi n}{n^{2}}$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 45

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$2,8,14,20, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:01

Problem 46

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, . . . .$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:08

Problem 47

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$-2,1,6,13,22, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
00:28

Problem 48

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \ldots$$

Linh Vu
Linh Vu
Numerade Educator
01:41

Problem 49

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$\frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \dots$$

Linh Vu
Linh Vu
Numerade Educator
01:01

Problem 50

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$2,24,720,40,320,3,628,800, \dots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 51

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$2,1+\frac{1}{2}, 1+\frac{1}{3}, 1+\frac{1}{4}, 1+\frac{1}{5}, \ldots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:02

Problem 52

Write an expression for the $n$ th term of the sequence. (There is more than one correct answer.)
$$\frac{1}{2 \cdot 3}, \frac{2}{3 \cdot 4}, \frac{3}{4 \cdot 5}, \frac{4}{5 \cdot 6}, \ldots$$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:52

Problem 53

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=4-\frac{1}{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:31

Problem 54

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\frac{3 n}{n+2}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:29

Problem 55

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=n e^{-n / 2}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:18

Problem 56

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\left(-\frac{2}{3}\right)^{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:24

Problem 57

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\left(\frac{2}{3}\right)^{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:21

Problem 58

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\left(\frac{3}{2}\right)^{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:22

Problem 59

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\sin \frac{n \pi}{6}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
02:38

Problem 60

Determine whether the sequence with the given $n$ th term is monotonic and whether it is bounded. Use a graphing utility to confirm your results.
$$a_{n}=\frac{\cos n}{n}$$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:32

Problem 61

(a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges, and (b) use a graphing utility to graph the first 10 terms of the =sequence and find its limit.
$$a_{n}=7+\frac{1}{n}$$

Nick Johnson
Nick Johnson
Numerade Educator
01:32

Problem 62

(a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges, and (b) use a graphing utility to graph the first 10 terms of the =sequence and find its limit.
$$a_{n}=5-\frac{2}{n}$$

Nick Johnson
Nick Johnson
Numerade Educator
01:40

Problem 63

(a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges, and (b) use a graphing utility to graph the first 10 terms of the =sequence and find its limit.
$$a_{n}=\frac{1}{3}\left(1-\frac{1}{3^{n}}\right)$$

Nick Johnson
Nick Johnson
Numerade Educator
01:32

Problem 64

(a) use Theorem 9.5 to show that the sequence with the given $n$ th term converges, and (b) use a graphing utility to graph the first 10 terms of the =sequence and find its limit.
$$a_{n}=2+\frac{1}{5^{n}}$$

Nick Johnson
Nick Johnson
Numerade Educator
00:47

Problem 65

Let $\left\{a_{n}\right\}$ be an increasing sequence such that $2 \leq a_{n} \leq 4 .$ Explain why $\left\{a_{n}\right\}$ has a limit. What can you conclude about the limit?

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
02:16

Problem 66

Let $\left\{a_{n}\right\}$ be a monotonic sequence such that $a_{n} \leq 1 .$ Discuss the convergence of $\left\{a_{n}\right\}$ When $\left\{a_{n}\right\}$ converges, what can you conclude about its limit?

Linh Vu
Linh Vu
Numerade Educator
01:29

Problem 67

Consider the sequence $\left\{A_{n}\right\}$ whose $n$ th term is given by $$A_{n}=P\left(1+\frac{r}{12}\right)^{n}$$
where $P$ is the principal, $A_{n}$ is the account balance after $n$ months, and $r$ is the interest rate compounded annually.
(a) Is $\left\{A_{n}\right\}$ a convergent sequence? Explain.
(b) Find the first 10 terms of the sequence when $P=\$ 10,000$ and $r=0.055$

Linh Vu
Linh Vu
Numerade Educator
02:14

Problem 68

A deposit of $\$ 100$ is made in an account at the beginning of each month at an annual interest rate of $3 \%$ compounded monthly. The balance in the account after $n$ months is $A_{n}=100(401)\left(1.0025^{n}-1\right)$.
(a) Compute the first six terms of the sequence $\left\{A_{n}\right\}$
(b) Find the balance in the account after 5 years by computing the 60 th term of the sequence.
(c) Find the balance in the account after 20 years by computing the 240 th term of the sequence.

Linh Vu
Linh Vu
Numerade Educator
01:11

Problem 69

Is it possible for a sequence to converge to two different numbers? If so, give an example. If not, explain why not.

Linh Vu
Linh Vu
Numerade Educator
01:57

Problem 70

In your own words, define each of the following.
(a) Sequence
(b) Convergence of a sequence
(c) Monotonic sequence
(d) Bounded sequence

Lauren Shelton
Lauren Shelton
Numerade Educator
01:44

Problem 71

Give an example of a sequence satisfying the condition or explain why no such sequence exists. (Examples are not unique.)
(a) A monotonically increasing sequence that converges to 10
(b) A monotonically increasing bounded sequence that does not converge
(c) A sequence that converges to $\frac{3}{4}$
(d) An unbounded sequence that converges to 100

Linh Vu
Linh Vu
Numerade Educator
00:32

Problem 72

The graphs of two sequences are shown in the figures. Which graph represents the sequence with alternating signs? Explain. (GRAPH CAN'T COPY)

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
04:54

Problem 73

A government program that currently costs taxpayers $\$ 4.5$ billion per year is cut back by 20 percent per year.
(a) Write an expression for the amount budgeted for this program after $n$ years.
(b) Compute the budgets for the first 4 years.
(c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator
01:29

Problem 74

When the rate of inflation is $4 \frac{1}{2} \%$ per year and the average price of a car is currently $\$ 25,000,$ the average price after $n$ years is $P_{n}=\$ 25,000(1.045)^{n} .$ Compute the average prices for the next 5 years.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
01:25

Problem 75

Compute the first six terms of the sequence $\left\{a_{n}\right\}=\{\sqrt[n]{n}\} .$ If the sequence converges, find its limit.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
01:45

Problem 76

Compute the first six terms of the sequence
$$\left\{a_{n}\right\}=\left\{\left(1+\frac{1}{n}\right)^{n}\right\}$$
If the sequence converges, find its limit.

Subhadeepta Sahoo
Subhadeepta Sahoo
Numerade Educator
02:05

Problem 77

Prove that if $\left\{s_{n}\right\}$ converges to $L$ and $L>0,$ then there exists a number $N$ such that $s_{n}>0$ for $n>N$.

Linh Vu
Linh Vu
Numerade Educator
03:06

Problem 78

The amounts of the federal debt $a_{n}$ (in trillions of dollars) of the United States from 2000 through 2011 are given below as ordered pairs of the form $\left(n, a_{n}\right)$ where $n$ represents the year, with $n=0$ corresponding to 2000. (Source: U.S. Office of Management and Budget) (0,5.6),(1,5.8),(2,6.2),(3,6.8),(4,7.4),(5,7.9),(6,8.5) (7,9.0),(8,10.0),(9,11.9),(10,13.5),(11,14.8)
(a) Use the regression capabilities of a graphing utility to find a model of the form $a_{n}=b n^{2}+c n+d, \quad n=0,1, \ldots, 11$ for the data. Use the graphing utility to plot the points and graph the model.
(b) Use the model to predict the amount of the federal debt in the year 2020

Regina Hays
Regina Hays
Numerade Educator
00:52

Problem 79

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges to 3 and $\left\{b_{n}\right\}$ converges to $2,$ then $\left\{a_{n}+b_{n}\right\}$ converges to 5.

Lauren Shelton
Lauren Shelton
Numerade Educator
01:32

Problem 80

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges, then $\lim _{n \rightarrow \infty}\left(a_{n}-a_{n+1}\right)=0$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:04

Problem 81

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ converges, then $\left\{a_{n} / n\right\}$ converges to $0 .$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:37

Problem 82

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left\{a_{n}\right\}$ diverges and $\left\{b_{n}\right\}$ diverges, then $\left\{a_{n}+b_{n}\right\}$ diverges.

Lauren Shelton
Lauren Shelton
Numerade Educator
07:12

Problem 83

In a study of the progeny of rabbits, Fibonacci (ca. $1170-$ ca. 1240 ) encountered the sequence now bearing his name. The sequence is defined recursively as $a_{n+2}=a_{n}+a_{n+1},$ where $a_{1}=1$ and $a_{2}=1$
(a) Write the first 12 terms of the sequence.
(b) Write the first 10 terms of the sequence defined by $b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1$
(c) Using the definition in part (b), show that $b_{n}=1+\frac{1}{b_{n-1}}$
(d) The golden ratio $\rho$ can be defined by $\lim _{n \rightarrow \infty} b_{n}=\rho .$ Show that $\rho=1+\frac{1}{\rho}$ and solve this equation for $\rho$

Mengchun Cai
Mengchun Cai
Numerade Educator
02:28

Problem 84

Show that the converse of Theorem 9.1 is not true. [Hint: Find a function $f(x)$ such that $f(n)=a_{n}$ converges, but $\left.\lim _{x \rightarrow \infty} f(x) \text { does not exist. }\right]$

Linh Vu
Linh Vu
Numerade Educator
03:00

Problem 85

Using a Sequence Consider the sequence $\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots$
(a) Compute the first five terms of this sequence.
(b) Write a recursion formula for $a_{n},$ for $n \geq 2$
(c) Find $\lim _{n \rightarrow \infty} a_{n}$

Lauren Shelton
Lauren Shelton
Numerade Educator
04:20

Problem 86

Consider the sequence $\left\{a_{n}\right\}$ where $a_{1}=\sqrt{k}, a_{n+1}=\sqrt{k+a_{n}},$ and $k>0$.
(a) Show that $\left\{a_{n}\right\}$ is increasing and bounded.
(b) Prove that $\lim _{n \rightarrow \infty} a_{n}$ exists.
(c) Find $\lim _{n \rightarrow \infty} a_{n^{*}}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:00

Problem 87

(a) Show that $\int_{1}^{n} \ln x d x<\ln (n !)$ for $n \geq 2$ (GRAPH CAN'T COPY)
(b) Draw a graph similar to the one above that shows $\ln (n !)<\int_{1}^{n+1} \ln x d x$
(c) Use the results of parts (a) and (b) to show that $\frac{n^{n}}{e^{n-1}}<n !<\frac{(n+1)^{n+1}}{e^{n}},$ for $n>1$
(d) Use the Squeeze Theorem for Sequences and the result of part (c) to show that $\lim _{n \rightarrow \infty}(\sqrt[n]{n !} / n)=1 / e$
(e) Test the result of part (d) for $n=20,50,$ and $100 .$

Hoan Nguyen
Hoan Nguyen
Numerade Educator
02:28

Problem 88

Prove, using the definition of the limit of a sequence, that $\lim _{n \rightarrow \infty} \frac{1}{n^{3}}=0$

Linh Vu
Linh Vu
Numerade Educator
01:43

Problem 89

Prove, using the definition of the limit of a sequence, that $\lim _{n \rightarrow \infty} r^{n}=0$ for $-1<r<1$.

Nick Johnson
Nick Johnson
Numerade Educator
01:08

Problem 90

Find a divergent sequence $\left\{a_{n}\right\}$ such that $\left\{a_{2 n}\right\}$ converges.

Linh Vu
Linh Vu
Numerade Educator
01:24

Problem 91

Prove Theorem 9.5 for a nonincreasing sequence.

Nick Johnson
Nick Johnson
Numerade Educator
01:22

Problem 92

Let $\left\{x_{n}\right\}, n \geq 0,$ be a sequence of nonzero real numbers such that $x_{n}^{2}-x_{n-1} x_{n+1}=1$ for $n=1,2,3, \ldots . .$ Prove there exists a real number $a$ such that $x_{n+1}=a x_{n}-x_{n-1}$ for all $n \geq 1$.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
05:41

Problem 93

Let $T_{0}=2, T_{1}=3, T_{2}=6,$ and for $n \geq 3$
$T_{n}=(n+4) T_{n-1}-4 n T_{n-2}+(4 n-8) T_{n-3}$
The first few terms are
2,3,6,14,40,152,784,5168,40,576
Find, with proof, a formula for $T_{n}$ of the form $T_{n}=A_{n}+B_{n}$ where $\left\{A_{n}\right\}$ and $\left\{B_{n}\right\}$ are well-known sequences.

Dorcas Attuabea Addo
Dorcas Attuabea Addo
Numerade Educator