Draw a picture to show that What can you conclude about the series?
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1
Functions and Models
1.1
Four Ways to Represent a Function
1.2
Mathematical Models: A Catalog of Essential Functions
1.3
New Functions from Old Functions
1.4
Graphing Calculators and Computers
1.5
Exponential Functions
1.6
Inverse Functions and Logarithms
2
Limits and Derivatives
2.1
The Tangent and Velocity Problems
2.2
The Limit of a Function
2.3
Calculating Limits Using the Limit Laws
2.4
The Precise Definition of a Limit
2.5
Continuity
2.6
Limits at Infinity; Horizontal Asymptotes
2.7
Tangents, Velocities, and Other Rates of Change
2.8
Derivatives
2.9
The Derivative as a Function
3
Differentiation Rules
3.1
Derivatives of Polynomials and Exponential Functions
3.10
Related Rates
3.11
Linear Approximations and Differentials
3.3
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3.4
Derivatives of Trigonometric Functions
3.5
The Chain Rule
3.6
Implicit Differentiation
3.7
Higher Derivatives
3.8
Derivatives of Logarithmic Functions
3.9
Hyperbolic Functions
4
Applications of Differentiation
4.1
Maximum and Minimum Values
4.10
Antiderivatives
4.2
The Mean Value Theorem
4.3
How Derivatives Affect the Shape of a Graph
4.4
Indeterminate Forms and LHospitals Rule
4.5
Summary of Curve Sketching
4.6
Graphing with Calculus and Calculators
4.7
Optimization Problems
4.8
Applications to Business and Economics
4.9
Newtons Method
5
Integrals
5.1
Areas and Distances
5.2
The Definite Integral
5.3
The Fundamental Theorem of Calculus
5.4
Indefinite Integrals and the Net Change Theorem
5.5
The Substitution Rule
5.6
The Logarithm Defined as an Integral
6
Applications of Integration
6.1
Areas between Curves
6.2
Volumes
6.3
Volumes by Cylindrical Shells
6.4
Work
6.5
Average Value of a Function
7
Techniques of Integration
7.1
Integration by Parts
7.2
Trigonometric Integrals
7.3
Trigonometric Substitution
7.4
Integration of Rational Functions by Partial Fractions
7.5
Strategy for Integration
7.6
Integration Using Tables and Computer Algebra Systems
7.7
Approximate Integration
7.8
Improper Integrals
8
Further Applications of Integration
8.1
Arc Length
8.2
Area of a Surface of Revolution
8.3
Applications to Physics and Engineering
8.4
Applications to Economics and Biology
8.5
Probability
9
Differential Equations
9.1
Modeling with Differential Equations
9.2
Direction Fields and Eulers Method
9.3
Separable Equations
9.4
Exponential Growth and Decay
9.5
The Logistic Equation
9.6
Linear Equations
9.7
Predator-Prey Systems
10
Parametric Equations and Polar Coordinates
10.1
Curves Defined by Parametric Equations
10.2
Calculus with Parametric Curves
10.3
Polar Coordinates
10.4
Areas and Lengths in Polar Coordinates
10.5
Conic Sections
10.6
Conic Sections in Polar Coordinates
11
Infinite Sequences and Series
11.1
Sequences
11.10
Taylor and Maclaurin Series
11.11
The Binomial Series
11.12
Applications of Taylor Polynomials
11.2
Series
11.3
The Integral Test and Estimates of Sums
11.4
The Comparison Tests
11.5
Alternating Series
11.6
Absolute Convergence and the Ratio and Root Tests
11.7
Strategy for Testing Series
11.8
Power Series
11.9
Representations of Functions as Power Series
12
Vectors and the Geometry of Space
12.1
Three-Dimensional Coordinate Systems
12.2
Vectors
12.3
The Dot Product
12.4
The Cross Product
12.5
Equations of Lines and Planes
12.6
Cylinders and Quadric Surfaces
12.7
Cylindrical and Spherical Coordinates
13
Vector Functions
13.1
Vector Functions and Space Curves
13.2
Derivatives and Integrals of Vector Functions
13.3
Arc Length and Curvature
13.4
Motion in Space: Velocity and Acceleration
14
Partial Derivatives
14.1
Functions of Several Variables
14.2
Limits and Continuity
14.3
Partial Derivatives
14.4
Tangent Planes and Linear Approximations
14.5
The Chain Rule
14.6
Directional Derivatives and the Gradient Vector
14.7
Maximum and Minimum Values
14.8
Lagrange Multipliers
15
Multiple Integrals
15.1
Double Integrals over Rectangles
15.2
Iterated Integrals
15.3
Double Integrals over General Regions
15.4
Double Integrals in Polar Coordinates
15.5
Applications of Double Integrals
15.6
Surface Area
15.7
Triple Integrals
15.8
Triple Integrals in Cylindrical and Spherical Coordinates
15.9
Change of Variables in Multiple Integrals
16
Vector Calculus
16.1
Vector Fields
16.2
Line Integrals
16.3
The Fundamental Theorem for Line Integrals
16.4
The Fundamental Theorem for Line Integrals
16.5
Curl and Divergence
16.6
Parametric Surfaces and Their Areas
16.7
Surface Integrals
16.8
Stokes Theorem
16.9
The Divergence Theorem
17
Second-Order Differential Equations
17.1
Second-Order Linear Equations
17.2
Nonhomogeneous Linear Equations
17.3
Applications of Second-Order Differential Equations
17.4
Series Solutions
Textbook Solutions for Calculus,
Chapter 11.3 Problem 31
Question
(a) Use the sum of the first 10 terms to estimate the sum of the series . How good is this estimate? (b) Improve this estimate using (3) with . (c) Find a value of that will ensure that the error in the approximation is less than .
Solution
The first step in solving 11.3 problem number 31 trying to solve the problem we have to refer to the textbook question: (a) Use the sum of the first 10 terms to estimate the sum of the series . How good is this estimate? (b) Improve this estimate using (3) with . (c) Find a value of that will ensure that the error in the approximation is less than .
From the textbook chapter The Integral Test and Estimates of Sums you will find a few key concepts needed to solve this.
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full solution
full solution
Title
Calculus, 5
Author
James Stewart
ISBN
9780534393397
Solved: (a) Use the sum of the first 10 terms to estimate the sum of the series . How
Chapter 11.3 textbook questions
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Chapter 11: Problem 1 Calculus, 5
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Chapter 11: Problem 2 Calculus, 5
Suppose is a continuous positive decreasing function for and . By drawing a picture, rank the following three quantities in increasing order:
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Chapter 11: Problem 3 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 4 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 5 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 6 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 7 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 8 Calculus, 5
Use the Integral Test to determine whether the series is convergent or divergent.
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Chapter 11: Problem 9 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 10 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 11 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 12 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 13 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 14 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 15 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 16 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 17 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 18 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 19 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 20 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 21 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 22 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 23 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 24 Calculus, 5
Determine whether the series is convergent or divergent.
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Chapter 11: Problem 29 Calculus, 5
The Riemann zeta-function is defined by and is used in number theory to study the distribution of prime numbers. What is the domain of ?
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Chapter 11: Problem 30 Calculus, 5
(a) Find the partial sum of the series . Estimate the error in using as an approximation to the sum of the series. (b) Use (3) with to give an improved estimate of the sum. (c) Find a value of so that is within of the sum.
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Chapter 11: Problem 31 Calculus, 5
(a) Use the sum of the first 10 terms to estimate the sum of the series . How good is this estimate? (b) Improve this estimate using (3) with . (c) Find a value of that will ensure that the error in the approximation is less than .
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Chapter 11: Problem 32 Calculus, 5
Find the sum of the series correct to three decimal places.
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Chapter 11: Problem 34 Calculus, 5
How many terms of the series would you need to add to find its sum to within ?
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Chapter 11: Problem 35 Calculus, 5
Show that if we want to approximate the sum of the series so that the error is less than 5 in the ninth decimal place, then we need to add more than terms!
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Chapter 11: Problem 36 Calculus, 5
(a) Show that the series is convergent. (b) Find an upper bound for the error in the approximation . (c) What is the smallest value of such that this upper bound is less than ? (d) Find for this value of .
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Chapter 11: Problem 37 Calculus, 5
(a) Use (4) to show that if is the partial sum of the harmonic series, then (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.
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Chapter 11: Problem 38 Calculus, 5
Use the following steps to show that the sequence has a limit. (The value of the limit is denoted by and is called Eulers constant.) (a) Draw a picture like Figure 6 with and interpret as an area [or use (5)] to show that for all . (b) Interpret as a difference of areas to show that . Therefore, is a decreasing sequence. (c) Use the Monotonic Sequence Theorem to show that is convergent.
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Chapter 11: Problem 39 Calculus, 5
Find all positive values of for which the series converges.
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