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CalculusMultivariable Calculus8th Edition

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Chapter 11.1, Problem 63E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 739 for advice on graphing sequences.)

63. a n = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ( 2 n − 1 ( 2 n ) n

To determine

Whether the sequence is convergent or divergent; guess the value of the limit if convergent with valid proof.

Explanation

Given:

The sequence is an=1⋅3⋅5⋅...⋅(2n−1)(2n)n .

Definition used:

If an is a sequence and limn→∞an exists, then the sequence an is said to be convergent; otherwise it is divergent.

Calculation:

Obtain the first 20 terms of the sequence.

n 1⋅3⋅5⋅...⋅(2n−1) (2n)n an=1⋅3⋅5⋅...⋅(2n−1)(2n)n
1 1 21=2 121=0.5000
2 3 42=16 342=0.1875
3 15 63=216 1563=0.0694
4 105 84=4,096 10584=0.0256
5 945 105=100,000 945105=0.0095
6 10,395 126=2,985,984 10,395126=0.0104
7 135,135 147=105,413,504 135,135147=0.0013
8 2,027,025 168=4,294,967,296 2,027,025168=0.0005
9 34,459,425 189=198,359,290,368 34,459,4259!=0.0002
10 654,729,075 2010=10,240,000,000,000 654,729,0752010=0.0001

Plot the points (n,an), for n=1,2,...10 on the graph as shown below in Figure 1.

From the graph, it is observed that the sequence is convergent.

Also, the value of the limit is guessed to 0 as the plotted points are closer to 0.

To prove:

The sequence an=1⋅3⋅5⋅...⋅(2n−1)(2n)n is convergent to 0.

Proof:

Obtain the limit of the sequence (the value of the term an as n tends to infinity)

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23-28 Use a linear approximation or differentials to estimate the given number. 100.5

Calculus (MindTap Course List)

Describe the distribution of sample means (shape, mean, standard error) for samples of n=64 selected from a pop...

Statistics for The Behavioral Sciences (MindTap Course List)

In Exercises 69-74, rationalize the numerator. 69. 2x3

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Evaluate the difference quotient for the given function. Simplify your answer. 30. f(x)=x+3x+1,f(x)f(1)x1

Single Variable Calculus

After all students have left the classroom, a statistics professor notices that four copies of the text were le...

Probability and Statistics for Engineering and the Sciences

Critical Thinking Results of a poll of a random sample of 3003 American adults showed that 20% did not know tha...

Understanding Basic Statistics

Thoughts Into Words Your friend keeps getting an answer of 30 when simplifying 7 82. What mistake is he making...

Intermediate Algebra

Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Finite Mathematics

Using the ordinary dating method, calculate the discount date and the net date for the following transactions. ...

Contemporary Mathematics for Business & Consumers

A sector with perimeter 30 in. has a bounding arc of length 12 in. Find the length of the radius of the circle.

Elementary Geometry For College Students, 7e

In Exercises 29 to 36, draw a Venn diagram to show each of the following sets. AB

Mathematical Excursions (MindTap Course List)

Fill in each blank: 45C=F

Elementary Technical Mathematics

31. Growth The Fish Commission states that the lengths of all fish in Spring Run are normally distributed, with...

Mathematical Applications for the Management, Life, and Social Sciences

Test the series for convergence or divergence. 2. 2325+2729+211

Single Variable Calculus: Early Transcendentals

Finding Values In Exercises 45-50 find the positive values of p for which the series converges. n=11n(lnn)[ln(l...

Calculus: Early Transcendental Functions (MindTap Course List)

Pumping Gasoline In Exercises 27 and 28, find the work done in pumping gasoline that weighs 42 pounds per cubic...

Calculus of a Single Variable

Distance and Bearing A man wandering in the desert walks 3.3 miles in the direction S 44 W. He then turns and w...

Trigonometry (MindTap Course List)

34. Let be the set of eight elements with identity element and noncommutative multiplication given by ...

Elements Of Modern Algebra

A definite integral for the length of y = x3, 1 x 2 is a) 121+x3dx b) 121+x6dx c) 121+9x4dx d) 121+3x2dx

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The unit vector in the direction of is:

Study Guide for Stewart's Multivariable Calculus, 8th

The Bureau of Alcohol, Tobacco, and Firearms (BATF) has been concerned about lead levels in California wines. I...

Introduction To Statistics And Data Analysis

Given: Acute 1 and AB Triangle ABC with A1, B1 and side AB

Elementary Geometry for College Students

In Exercise 33 and 34, use the above table. BLOOD TYPES a. What is the probability that a person selected rando...

Finite Mathematics for the Managerial, Life, and Social Sciences

Orthogonal Vectors In Exercises 51-54, find two vectors in opposite directions that are orthogonal to the vecto...

Multivariable Calculus

Finding a Point of IntersectionIn Exercises 3336, determine whether the lines intersect, and if so, find the po...

Calculus (MindTap Course List)

Tangent Planes of a ConeShow that any tangent plane to the cone z2=a2x2+b2y2 passes through the origin.

Calculus

A pair of dice is rolled. Using ordered pairs, give the following. a. The sample space. HINT:S has 36 elements,...

Mathematics: A Practical Odyssey

Define an operational definition and explain the purpose and the limitations of operational definitions.

Research Methods for the Behavioral Sciences (MindTap Course List)

Read the setting of these metric micrometer scales graduated in 0.01 mm.

Mathematics For Machine Technology

Define a positive relationship and a negative relationship and explain the how consistency of positive and nega...

Research Methods for the Behavioral Sciences (MindTap Course List)

reconsider the observed frequencies in exercise 1 a. Compute the sample proportion for each population. b. use ...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

The following table shows the total return and the number of funds for four categories of mutual funds. Type of...

Statistics for Business & Economics, Revised (MindTap Course List)

Each function is a stretching offx=x3. Graph both on one set of coordinate axes. gx=13x3

College Algebra (MindTap Course List)

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