Graphical Analysis The figures show the graphs of the first 10 terms, and the graphs of the first 10 terms of the sequence of partial sums, of each series. and (a) Identify the series in each figure. (b) Which series is a series? Does it converge or diverge? (c) For the series that are not series, how do the magnitudes of the terms compare with the magnitudes of the terms of the series? What conclusion can you draw about the convergence or divergence of the series? (d) Explain the relationship between the magnitudes of the terms of the series and the magnitudes of the terms of the partial sums.
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P
Preparation for Calculus
P.1
Graphs and Models
P.2
Linear Models and Rates of Change
P.3
Functions and Their Graphs
P.4
Fitting Models to Data
1
Limits and Their Properties
1.1
A Preview of Calculus
1.2
Finding Limits Graphically and Numerically
1.3
Evaluating Limits Analytically
1.4
Continuity and One-Sided Limits
1.5
Infinite Limits
2
Differentiation
2.1
The Derivative and the Tangent Line Problem
2.2
Basic Differentiation Rules and Rates of Change
2.3
Product and Quotient Rules and Higher-Order Derivatives
2.4
The Chain Rule
2.5
Implicit Differentiation
2.6
Related Rates
3
Applications of Differentiation
3.1
Extrema on an Interval
3.2
Rolles Theorem and the Mean Value Theorem
3.3
Increasing and Decreasing Functions and the First Derivative Test
3.4
Concavity and the Second Derivative Test
3.5
Limits at Infinity
3.6
A Summary of Curve Sketching
3.7
Optimization Problems
3.8
Newtons Method
3.9
Differentials
4
Integration
4.1
Antiderivatives and Indefinite Integration
4.2
Area
4.3
Riemann Sums and Definite Integrals
4.4
The Fundamental Theorem of Calculus
4.5
Integration by Substitution
4.6
Numerical Integration
5
Logarithmic, Exponential, and Other Transcendental Functions
5.1
The Natural Logarithmic Function: Differentiation
5.2
The Natural Logarithmic Function: Integration
5.3
Inverse Functions
5.4
Exponential Functions: Differentiation and Integration
5.5
Bases Other Than e and Applications
5.6
Inverse Trigonometric Functions: Differentiation
5.7
Inverse Trigonometric Functions: Integration
5.8
Hyperbolic Functions
6
Differential Equations
6.1
Slope Fields and Eulers Method
6.2
Differential Equations: Growth and Decay
6.3
Separation of Variables and the Logistic Equation
6.4
First-Order Linear Differential Equations
7
Applications of Integration
7.1
Area of a Region Between Two Curves
7.2
Volume: The Disk Method
7.3
Volume: The Shell Method
7.4
Arc Length and Surfaces of Revolution
7.5
Work
7.6
Moments, Centers of Mass, and Centroids
7.7
Fluid Pressure and Fluid Force
8
Integration Techniques, LHpitals Rule, and Improper Integrals
8.1
Basic Integration Rules
8.2
Integration by Parts
8.3
Trigonometric Integrals
8.4
Trigonometric Substitution
8.5
Partial Fractions
8.6
Integration by Tables and Other Integration Techniques
8.7
Indeterminate Forms and LHpitals Rule
8.8
Improper Integrals
9
Infinite Series
9.1
Sequences
9.10
Taylor and Maclaurin Series
9.2
Series and Convergence
9.3
The Integral Test and p-Series
9.4
Comparisons of Series
9.5
Alternating Series
9.6
The Ratio and Root Tests
9.7
Taylor Polynomials and Approximations
9.8
Power Series
9.9
Representation of Functions by Power Series
10
Conics, Parametric Equations, and Polar Coordinates
10.1
Conics and Calculus
10.2
Plane Curves and Parametric Equations
10.3
Parametric Equations and Calculus
10.4
Polar Coordinates and Polar Graphs
10.5
Area and Arc Length in Polar Coordinates
10.6
Polar Equations of Conics and Keplers Laws
11
Vectors and the Geometry of Space
11.1
Vectors in the Plane
11.2
Space Coordinates and Vectors in Space
11.3
The Dot Product of Two Vectors
11.4
The Cross Product of Two Vectors in Space
11.5
Lines and Planes in Space
11.6
Surfaces in Space
11.7
Cylindrical and Spherical Coordinates
12
Vector-Valued Functions
12.1
Vector-Valued Functions
12.2
Differentiation and Integration of Vector-Valued Functions
12.3
Velocity and Acceleration
12.4
Tangent Vectors and Normal Vectors
12.5
Arc Length and Curvature
13
Functions of Several Variables
13.1
Introduction to Functions of Several Variables
13.10
Lagrange Multipliers
13.2
Limits and Continuity
13.3
Partial Derivatives
13.4
Differentials
13.5
Chain Rules for Functions of Several Variables
13.6
Directional Derivatives and Gradients
13.7
Tangent Planes and Normal Lines
13.8
Extrema of Functions of Two Variables
13.9
Applications of Extrema of Functions of Two Variables
14
Multiple Integration
14.1
Iterated Integrals and Area in the Plane
14.2
Double Integrals and Volume
14.3
Change of Variables: Polar Coordinates
14.4
Center of Mass and Moments of Inertia
14.5
Surface Area
14.6
Triple Integrals and Applications
14.7
Triple Integrals in Cylindrical and Spherical Coordinates
14.8
Change of Variables: Jacobians
15
Vector Analysis
15.1
Vector Fields
15.2
Line Integrals
15.3
Conservative Vector Fields and Independence of Path
15.4
Greens Theorem
15.5
Parametric Surfaces
15.6
Surface Integrals
15.7
Divergence Theorem
15.8
Stokess Theorem
Textbook Solutions for Calculus
Chapter 9.4 Problem 29
Question
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
Solution
The first step in solving 9.4 problem number 29 trying to solve the problem we have to refer to the textbook question: In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
From the textbook chapter Comparisons of Series you will find a few key concepts needed to solve this.
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Title
Calculus 8
Author
Ron Larson Robert P. Hostetler, Bruce H. Edwards
ISBN
9780618502981
In Exercises 2936, test for convergence or divergence, using each test at least once
Chapter 9.4 textbook questions
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Chapter 9: Problem 1 Calculus 8
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Chapter 9: Problem 2 Calculus 8
Graphical Analysis The figures show the graphs of the first 10 terms, and the graphs of the first 10 terms of the sequence of partial sums, of each series. and (a) Identify the series in each figure. (b) Which series is a series? Does it converge or diverge? (c) For the series that are not series, how do the magnitudes of the terms compare with the magnitudes of the terms of the series? What conclusion can you draw about the convergence or divergence of the series? (d) Explain the relationship between the magnitudes of the terms of the series and the magnitudes of the terms of the partial sums.
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Chapter 9: Problem 3 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 4 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 5 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 6 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 7 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 8 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 9 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 10 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 11 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 12 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 13 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 14 Calculus 8
In Exercises 314, use the Direct Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 15 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 16 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 17 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 18 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 19 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 20 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 21 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 22 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 23 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 24 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 25 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 26 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 27 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 28 Calculus 8
In Exercises 1528, use the Limit Comparison Test to determine the convergence or divergence of the series.
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Chapter 9: Problem 29 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 30 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 31 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 32 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 33 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 34 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 35 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 36 Calculus 8
In Exercises 2936, test for convergence or divergence, using each test at least once. Identify which test was used. (a) Term Test (b) Geometric Series Test (c) Series Test (d) Telescoping Series Test (e) Integral Test (f) Direct Comparison Test (g) Limit Comparison Test
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Chapter 9: Problem 37 Calculus 8
Use the Limit Comparison Test with the harmonic series to show that the series (where ) diverges if is finite and nonzero.
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Chapter 9: Problem 38 Calculus 8
Prove that, if and are polynomials of degree and respectively, then the series converges if and diverges if
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Chapter 9: Problem 39 Calculus 8
In Exercises 3942, use the polynomial test given in Exercise 38 to determine whether the series converges or diverges.
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Chapter 9: Problem 40 Calculus 8
In Exercises 3942, use the polynomial test given in Exercise 38 to determine whether the series converges or diverges.
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Chapter 9: Problem 41 Calculus 8
In Exercises 3942, use the polynomial test given in Exercise 38 to determine whether the series converges or diverges.
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Chapter 9: Problem 42 Calculus 8
In Exercises 3942, use the polynomial test given in Exercise 38 to determine whether the series converges or diverges.
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Chapter 9: Problem 43 Calculus 8
In Exercises 43 and 44, use the divergence test given in Exercise 37 to show that the series diverges.
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Chapter 9: Problem 44 Calculus 8
In Exercises 43 and 44, use the divergence test given in Exercise 37 to show that the series diverges.
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Chapter 9: Problem 45 Calculus 8
In Exercises 4548, determine the convergence or divergence of the series.
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Chapter 9: Problem 46 Calculus 8
In Exercises 4548, determine the convergence or divergence of the series.
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Chapter 9: Problem 47 Calculus 8
In Exercises 4548, determine the convergence or divergence of the series.
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Chapter 9: Problem 48 Calculus 8
In Exercises 4548, determine the convergence or divergence of the series.
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Chapter 9: Problem 49 Calculus 8
Review the results of Exercises 4548. Explain why careful analysis is required to determine the convergence or divergence of a series and why only considering the magnitudes of the terms of a series could be misleading.
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Chapter 9: Problem 50 Calculus 8
State the Direct Comparison Test and give an example of its use.
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Chapter 9: Problem 51 Calculus 8
State the Limit Comparison Test and give an example of its use.
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Chapter 9: Problem 52 Calculus 8
It appears that the terms of the series are less than the corresponding terms of the convergent series If the statement above is correct, the first series converges. Is this correct? Why or why not? Make a statement about how the divergence or convergence of a series is affected by inclusion or exclusion of the first finite number of terms.
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Chapter 9: Problem 53 Calculus 8
The figure shows the first 20 terms of the convergent series and the first 20 terms of the series Identify the two series and explain your reasoning in making the selection.
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Chapter 9: Problem 54 Calculus 8
Consider the series (a) Verify that the series converges. (b) Use a graphing utility to complete the table. (c) The sum of the series is Find the sum of the series (d) Use a graphing utility to find the sum of the series
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Chapter 9: Problem 55 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 56 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 57 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 58 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 59 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 60 Calculus 8
True or False? In Exercises 5560, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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Chapter 9: Problem 61 Calculus 8
Prove that if the nonnegative series and converge, then so does the series
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Chapter 9: Problem 62 Calculus 8
Use the result of Exercise 61 to prove that if the nonnegative series converges, then so does the series
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Chapter 9: Problem 63 Calculus 8
Find two series that demonstrate the result of Exercise 61.
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Chapter 9: Problem 64 Calculus 8
Find two series that demonstrate the result of Exercise 62.
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Chapter 9: Problem 65 Calculus 8
Suppose that and are series with positive terms. Prove that if and converges, also converges.
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Chapter 9: Problem 66 Calculus 8
Suppose that and are series with positive terms. Prove that if and diverges, also diverges.
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Chapter 9: Problem 67 Calculus 8
Use the result of Exercise 65 to show that each series converges.
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Chapter 9: Problem 68 Calculus 8
Use the result of Exercise 66 to show that each series diverges.
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Chapter 9: Problem 69 Calculus 8
Suppose that is a series with positive terms. Prove that if converges, then also converges.
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Chapter 9: Problem 72 Calculus 8
Prove that if is a convergent series of positive real numbers, then so is
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