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CalculusCalculus: Early Transcendental Functions (MindTap Course List)Using Power Series (a) Consider the power series ∑ n = 0 ∞ a n x n = 1 + 2 x + 3 x 2 + x 3 + 2 x 4 + 3 x 5 + x 6 + ⋯ in which the coefficients a n = 1 , 2 , 3 , 1 , 2 , 3 , 1 , … are periodic of period p = 3 . Find the radius of convergence and the sum of this power series. (b) Consider a power series ∑ n = 0 ∞ a n x n in which the coefficients are periodic, ( a n + p = a p ) , and a n > 0 . Find the radius of convergence and the sum of this power series.BuyFind*arrow_forward*

6th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285774770

Chapter 9, Problem 6PS

Textbook Problem

Using Power Series

(a) Consider the power series

in which the coefficients

(b) Consider a power series

in which the coefficients are periodic,

Calculus: Early Transcendental Functions (MindTap Course List)

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Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 5-10,...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Writing the Terms of a Sequence In Exercises 11...Ch. 9.1 - Matching In Exercises 13-16, match the sequence...Ch. 9.1 - Matching In Exercises 13-16, match the sequence...

Ch. 9.1 - Matching In Exercises 13-16, match the sequence...Ch. 9.1 - Matching In Exercises 13-16, match the sequence...Ch. 9.1 -
Writing Terms In Exercises 13–16. write the next...Ch. 9.1 - Writing Terms In Exercises 1316. write the next...Ch. 9.1 - Writing Terms In Exercises 1316. write the next...Ch. 9.1 - Writing Terms In Exercises 1316. write the next...Ch. 9.1 - Simplifying Factorials In Exercises 1720, simplify...Ch. 9.1 -
Simplifying Factorials In Exercises 17–20,...Ch. 9.1 - Simplifying Factorials In Exercises 1720, simplify...Ch. 9.1 - Simplifying Factorials In Exercises 1720, simplify...Ch. 9.1 - Finding the Limit of a Sequence In Exercises 2124,...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Finding the Limit of a Sequence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Determining Convergence or Divergence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Finding the nth Term of a Sequence In Exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Monotonic and Bounded Sequences In exercises...Ch. 9.1 - Using a Theorem In Exercises 61-64, (a) use...Ch. 9.1 - Using a Theorem In Exercises 61-64, (a) use...Ch. 9.1 - Using a Theorem In Exercises 61-64, (a) use...Ch. 9.1 - Using a Theorem In Exercises 61-64, (a) use...Ch. 9.1 - Increasing Sequence Let {an} be an increasing...Ch. 9.1 - Compound Interest Consider the sequence An whose...Ch. 9.1 - Compound Interest A deposit of $100 is made in an...Ch. 9.1 - Inflation When the rate of inflation is 412 per...Ch. 9.1 - EXPLORING CONCEPTS Monotonic Sequence Let an be a...Ch. 9.1 -
69. Sequence Is it possible for a sequence to...Ch. 9.1 -
70. Defining Terms In your own words, define each...Ch. 9.1 - Writing a Sequence Give at example of a sequence...Ch. 9.1 - HOW DO YOU SEE IT? The graphs of two sequences are...Ch. 9.1 -
73. Government Expenditures A government program...Ch. 9.1 - Using a Sequence Compute the first six terms of...Ch. 9.1 - Using a Sequence Compute the first six terms of...Ch. 9.1 - Proof Prove that if sn converges to L and L0, then...Ch. 9.1 - Modeling DataThe amounts of the federal debt an...Ch. 9.1 - True or False? In Exercises 77-80, determine...Ch. 9.1 - True or False? In Exercises 77-80, determine...Ch. 9.1 - True or False? In Exercises 77-80, determine...Ch. 9.1 - True or False? In Exercises 77-80, determine...Ch. 9.1 - Fibonacci Sequence In a study of the progeny of...Ch. 9.1 - Using a Theorem Show that the converse of Theorem...Ch. 9.1 - Using a Sequence Consider dis sequence...Ch. 9.1 - Using a Sequence Consider die sequence an where...Ch. 9.1 - Squeeze Theorem (a) Show that 1nlnxdxln(n!) for n2...Ch. 9.1 - Proof Prove, using the definition of the limit of...Ch. 9.1 - Proof Prove, using die definition of the limit of...Ch. 9.1 - Using a Sequence Find a divergent sequence an such...Ch. 9.1 - Proof Prove Theorem 9.5 for a nonincreasing...Ch. 9.1 - Let xn,n0, be a sequence of nonzero real numbers...Ch. 9.1 - T0=2,T1=3,T2=6, and for n3...Ch. 9.2 - Sequence and Series Describe the difference...Ch. 9.2 - Geometric Series Define a geometric series, state...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Finding Partial Sums In Exercises 5 10, find the...Ch. 9.2 - Verifying Divergence In Exercises 714, verify that...Ch. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Verifying Divergence In Exercises 11-18, verify...Ch. 9.2 - Verifying Divergence In Exercises 714, verify that...Ch. 9.2 - Verifying Divergence In Exercises 714, verify that...Ch. 9.2 - Verifying Divergence In Exercises 714, verify that...Ch. 9.2 - Verifying Divergence In Exercises 714, verify that...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercises 1520, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Verifying Convergence In Exercise 19-24, verify...Ch. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Finding the Sum of a Convergent Series In...Ch. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Using a Geometric Series In Exercises 3540, (a)...Ch. 9.2 - Using a Geometric Series In Exercises 3540, (a)...Ch. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Using a Geometric Series In Exercises 39-44, (a)...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Determining Convergence or Divergence In Exercises...Ch. 9.2 - Using a Series (a) You delete a finite number of...Ch. 9.2 - Series Slate the definitions of convergent and...Ch. 9.2 - nth-Term Test for Divergence State the nth-Term...Ch. 9.2 - Comparing SeriesExplain any differences among the...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Making a Series Converge In Exercises 61-66, find...Ch. 9.2 - Using a Geometric Series In Exercise 67 and 68,...Ch. 9.2 - Using a Geometric Series In Exercise 67 and 68,...Ch. 9.2 - Writing In Exercises 69 and 70, use graphing...Ch. 9.2 - Writing In Exercises 69 and 70, use graphing...Ch. 9.2 - Marketing An electronic games manufacturer...Ch. 9.2 - Depreciation A company buys a machine for $475,000...Ch. 9.2 - Multiplier Effect The total annual spending by...Ch. 9.2 - Multiplier Effect Repeat Exercise 73 when the...Ch. 9.2 - Distance A ball is dropped from a height of 16...Ch. 9.2 - Time The ball in Exercise 75 takes the following...Ch. 9.2 - Probability In Exercises 77 and 78. the random...Ch. 9.2 - Probability In Exercises 77 and 78, the random...Ch. 9.2 - Probability A fair coin is tossed repeatedly. The...Ch. 9.2 - Probability In an experiment, three people toss a...Ch. 9.2 - Area The sides of a square are 16 inches in...Ch. 9.2 - Length A right triangle XYZ is shown above where...Ch. 9.2 - Using a Geometric Series In Exercises 83-86, use...Ch. 9.2 - Annuities When an employee receives a paycheck at...Ch. 9.2 - Salary You go to work at a company that pays $0.01...Ch. 9.2 - Sphereflake The sphereflakeshown below is a...Ch. 9.2 - Annuities In Exercises 8790, consider making...Ch. 9.2 - Annuities In Exercises 8790, consider making...Ch. 9.2 - Annuities In Exercises 8790, consider making...Ch. 9.2 - Annuities In Exercises 8790, consider making...Ch. 9.2 - True or False? In Exercises 91-96, determine...Ch. 9.2 - True or False? In Exercises 91-96, determine...Ch. 9.2 - True or False? In Exercises 91-96. determine...Ch. 9.2 - True or False? In Exercises 91-96, determine...Ch. 9.2 - True or False? In Exercises 91-96, determine...Ch. 9.2 - True or False? In Exercises 91-96, determine...Ch. 9.2 - Using Divergent Series Find two divergent series...Ch. 9.2 - Proof Given two infinite series an and bn such...Ch. 9.2 - Fibonacci Sequence The Fibonacci sequence is...Ch. 9.2 - Remainder Let an be a convergent series, and let...Ch. 9.2 - Proof Prove that 1r+1r2+1r3+=1r1, for r1.Ch. 9.2 - HOW DO YOU SEE IT? The figure below represents an...Ch. 9.2 - Express k=16k(3k+12k1)(3k2k) as a rational...Ch. 9.2 - Let be the sum of the first n terms of the...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 122, confirm...Ch. 9.3 - Using the Integral Test In Exercises 122, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 122, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 3-22, confirm...Ch. 9.3 - Using the Integral Test In Exercises 23 and 24,...Ch. 9.3 - Using the Integral Test In Exercises 23 and 24,...Ch. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Conditions of the Integral Test In Exercises...Ch. 9.3 - Using the Integral Test In Exercises 2932, use the...Ch. 9.3 - Using the Integral Test In Exercises 29-32, use...Ch. 9.3 - Using the Integral Test In Exercises 29-32, use...Ch. 9.3 - Using the Integral Test In Exercises 29-32, use...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 3338, use Theorem...Ch. 9.3 - Using a p-Series In Exercises 33-38, use Theorem...Ch. 9.3 - Numerical and Graphical Analysis Use a graphing...Ch. 9.3 - Numerical Reasoning Because the harmonic series...Ch. 9.3 - Integral TestState the Integral Test and give an...Ch. 9.3 - p-Series Define a p-series and state the...Ch. 9.3 - EXPLORING CONCEPTS Think About It Without...Ch. 9.3 - EXPLORING CONCEPTS Using a Function Let f be a...Ch. 9.3 - Using a Series Use a graph lo show that the...Ch. 9.3 - HOW DO YOU SEE IT? The graphs show the sequences...Ch. 9.3 - Finding Values In Exercises 45-50, find the...Ch. 9.3 - Finding Values In Exercises 45-50. find the...Ch. 9.3 - Finding Values In Exercises 45-50, find the...Ch. 9.3 - Finding Values In Exercises 45-50 find the...Ch. 9.3 - Finding Values In Exercises 45-50, find the...Ch. 9.3 - Finding Values In Exercises 45-50 find the...Ch. 9.3 - Proof Let f be a positive, continuous, and...Ch. 9.3 - Using a Remainder Show that the result of Exercise...Ch. 9.3 - Approximating a Sum In Exercises 5560, use the...Ch. 9.3 - Approximating a Sum In Exercises 5560, use the...Ch. 9.3 - Approximating a Sum In Exercises 53-58, use the...Ch. 9.3 - Approximating a Sum In Exercises 53-58, use the...Ch. 9.3 - Approximating a Sum In Exercises 53-58, use the...Ch. 9.3 - Approximating a Sum In Exercises 5560, use the...Ch. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Finding a Value In Exercises 59-62, use the result...Ch. 9.3 - Comparing Series (a) Show that n=21n1,1 converges...Ch. 9.3 - Using a p-Series Ten terms arc used to approximate...Ch. 9.3 - Eulers Constant Let Sn=k=1n1k=1+12++1n (a) Show...Ch. 9.3 - Finding a Sum Find the sum of the series...Ch. 9.3 - Using a Series Consider the series n=2xlnn (a)...Ch. 9.3 - Riemann Zeta Function The Riemann zeta function...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 7182, determine the...Ch. 9.3 - Review In Exercises 7182, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 7180, test for convergence or...Ch. 9.3 - Review In Exercises 69-80, determine the...Ch. 9.3 - Review In Exercises 7182, determine the...Ch. 9.4 - Graphical Analysis In Exercises 3 and 4, the...Ch. 9.4 - Graphical Analysis In Exercises 3 and 4, the...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exersices5-16,...Ch. 9.4 - Using the Direct Comparison Test In Exersices5-16,...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises...Ch. 9.4 - Using the Direct Comparison Test In Exercises 312,...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - n=11n2(n+3)Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Using the Limit Comparison Test In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Using the Limit Comparison Test Use the Limit...Ch. 9.4 - Proof Prove that, if P(n) and Q(n) are polynomials...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - n=1n2n3+1Ch. 9.4 - Verifying Divergence In Exercises 41 and 42, use...Ch. 9.4 - Verifying Divergence In Exercises 41 and 42, use...Ch. 9.4 - Determining Convergence or Divergence In Exercises...Ch. 9.4 - Determining Convergence or DivergenceIn Exercises...Ch. 9.4 - Determining Convergence or Divergence In Exorcises...Ch. 9.4 - Determining Convergence or Divergence In Exorcises...Ch. 9.4 - Using Series Review the results of Exercises 43-46...Ch. 9.4 - Comparing Series It appears that the series...Ch. 9.4 - Limit Comparison Test State the Limit Comparison...Ch. 9.4 - Direct Comparison TestState the Direct Comparison...Ch. 9.4 - Using a Series Consider the series n=11(2n1)2 (a)...Ch. 9.4 - HOW DO YOU SEE IT? The figure shows the first 20...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - True or False? In Exercises 53-58, determine...Ch. 9.4 - Proof Prove that if the nonnegative series n=1an...Ch. 9.4 - Proof Use the result of Exercise 59 to prove that...Ch. 9.4 - Finding Series Find two series that demonstrate...Ch. 9.4 - Finding Series Find two scries that demonstrate...Ch. 9.4 - Proof Suppose that an and bn are series with...Ch. 9.4 - Proof Suppose that an and bn are series with...Ch. 9.4 - Verifying ConvergenceUse the result of Exercise 59...Ch. 9.4 - Verifying DivergenceUse the result of Exercise 60...Ch. 9.4 - Proof Suppose that an is a series with positive...Ch. 9.4 - Proof Prove that the series n=111+2+3++n...Ch. 9.4 - Comparing Series Show that n=1lnnnn converges by...Ch. 9.4 - PUTNAM EXAM CHALLENGE Is the infinite series...Ch. 9.4 - PUTNAM EXAM CHALLENGE Prove that if n=1an is a...Ch. 9.5 - Absolute and Conditional Convergence In your own...Ch. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Numerical and Graphical Analysis In Exercises 5-8,...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Determining Convergence or Divergence In Exercises...Ch. 9.5 - Approximating the Sum of an Alternating Series In...Ch. 9.5 - Approximating the Sum of an Alternating Series In...Ch. 9.5 - Approximating the Sum of an Alternating Series In...Ch. 9.5 - Approximating the Sum of an Alternating Series In...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Finding the Number of Terms In Exercises 35-40,...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - Alternating Series Define an alternating series.Ch. 9.5 - Alternating Series TestState the Alternating...Ch. 9.5 - Alternating Series RemainderGive the remainder...Ch. 9.5 - Determining Absolute and Conditional Convergence...Ch. 9.5 - HOW DO YOU SEE IT? The graphs of the sequences of...Ch. 9.5 - True or False? In Exercises 61 and 62, determine...Ch. 9.5 - True or False? In Exercises 61 and 62, determine...Ch. 9.5 - Finding Values In Exercises 63 and 64, find the...Ch. 9.5 - Finding Values In Exercises 63 and 64, find the...Ch. 9.5 - Proof Prove that if an converges, then an2...Ch. 9.5 - Finding a Series Use the result of Exercise 63 to...Ch. 9.5 - Finding a Series Give an example of a series that...Ch. 9.5 - Finding Values Find all values of x for which the...Ch. 9.5 - Using a Series In Exercises 69 and 70, use the...Ch. 9.5 - Using a Series In Exercises 69 and 70, use the...Ch. 9.5 - Review In Exercises 7180, test for convergence or...Ch. 9.5 - Review In Exercises 7180, test for convergence or...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 7180, test for convergence or...Ch. 9.5 - Review In Exercises 7180, test for convergence or...Ch. 9.5 - Review In Exercises 7180, test for convergence or...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Review In Exercises 71-80, determine the...Ch. 9.5 - Describing an Error The following argument that...Ch. 9.5 - PUTNAM EXAM CHALLENGE Assume as known the (true)...Ch. 9.6 - Think About ItWhat can you conclude about the...Ch. 9.6 - Verifying a Formula In Exercises 14, verify the...Ch. 9.6 - Verifying a Formula In Exercises 7 and 8, verify...Ch. 9.6 - Verifying a Formula In Exercises 14, verify the...Ch. 9.6 - Verifying a Formula In Exercises 14, verify the...Ch. 9.6 - MatchingIn Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Matching In Exercises 9-14, match the series with...Ch. 9.6 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.6 - Numerical, Graphical, and Analytic Analysis In...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio Test In Exercises 1334, use the...Ch. 9.6 - Using the Ratio TestIn Exercises 1334, use the...Ch. 9.6 - Using the Ratio TestIn Exercises 1334, use the...Ch. 9.6 - Using the Root Test In Exercises 3550, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 3550, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Using the Root Test In Exercises 39-52, use the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Review In Exercises 53-70, determine the...Ch. 9.6 - Identifying Series In Exercises 71-74, identify...Ch. 9.6 - Identifying Series In Exercises 71-74, identify...Ch. 9.6 - Identifying Series In Exercise 71-74, identify the...Ch. 9.6 - Identifying Series In Exercise 71-74, identify the...Ch. 9.6 - Writing an Equivalent Series In Exercises 75 and...Ch. 9.6 - Writing an Equivalent Series In Exercises 73 and...Ch. 9.6 - Finding the Number of Terms In Exercises 75 and...Ch. 9.6 - Finding the Number of Terms In Exercises 75 and...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using a Recursively Defined Series In Exercises...Ch. 9.6 - Using the Ratio Test or Root Test In Exercise...Ch. 9.6 - Using the Ratio Test or Root Test In Exercise...Ch. 9.6 - Using the Ratio Test or Root Test In Exercise...Ch. 9.6 - Using the Ratio Test or Root Test In Exercise...Ch. 9.6 - Finding Values In Exercises 87-92, find the values...Ch. 9.6 - Finding Values In Exercise 87-92, find the values...Ch. 9.6 - Finding Values In Exercise 87-92, find the values...Ch. 9.6 - Finding Values In Exercises 87-92, find the values...Ch. 9.6 - Finding Values In Exercise 87-92, find the values...Ch. 9.6 - Finding Values In Exercise 87-92, find the values...Ch. 9.6 - WRITING ABOUT CONCEPTS Ratio Test State the Ratio...Ch. 9.6 - WRITING ABOUT CONCEPTS Root Test State the Root...Ch. 9.6 - Think About It You are told that the terms of a...Ch. 9.6 - WRITING ABOUT CONCEPTS Using an Alternating Series...Ch. 9.6 - HOW DO YOU SEE IT?The figure shows the first 10...Ch. 9.6 - Proof Prove Property 2 of Theorem 9.17.Ch. 9.6 - Proof Prove Theorem 9.18. (Hint for Property I: If...Ch. 9.6 - Verifying an Inconclusive Test In Exercise 99-102,...Ch. 9.6 - Verifying an Inconclusive Test In Exercises...Ch. 9.6 - Verifying an Inconclusive Test In Exercise 99-102,...Ch. 9.6 - Verifying an Inconclusive Test In Exercise 99-102,...Ch. 9.6 - Verifying an Inconclusive Test Show that the Root...Ch. 9.6 - Verifying Inconclusive Tests Show that the Ratio...Ch. 9.6 - Using Values Determine the convergence or...Ch. 9.6 - Using a Series Show this if n=1an is absolutely...Ch. 9.6 - PUTNAM EXAM CHALLENGE Show that if the series...Ch. 9.6 - PUTNAM EXAM CHALLENGE 51 Is the following series...Ch. 9.7 - CONCEPT CHECK Polynomial Approximation An...Ch. 9.7 - CONCEPT CHECK, Accuracy of a Taylor Polynomial...Ch. 9.7 - CONCEPT CHECK, Accuracy of a Taylor Polynomial In...Ch. 9.7 - Matching In Exercises 5-8, match the Taylor...Ch. 9.7 - Matching In Exercises 5-8, match the Taylor...Ch. 9.7 - Matching In Exercises 5-8, match the Taylor...Ch. 9.7 - Matching In Exercises 5-8, match the Taylor...Ch. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Finding a First-Degree Polynomial ApproximationIn...Ch. 9.7 - Finding a First-Degree Polynomial Approximation In...Ch. 9.7 - Graphical and Numerical Analysis In Exercises 13...Ch. 9.7 - Graphical and Numerical Analysis In Exercises 13...Ch. 9.7 - Conjecture Consider the function f(x)=cosx and its...Ch. 9.7 - Conjecture Consider the function f(x)=x2ex. (a)...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 1324,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 1324,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 1324,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Maclaurin Polynomial In Exercises 17-26,...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding a Taylor Polynomial In Exercises 2530,...Ch. 9.7 - Finding a Taylor Polynomial InExercises 2530, find...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding a Taylor PolynomialIn Exercises 27-32,...Ch. 9.7 - Finding Taylor Polynomials Using Technology In...Ch. 9.7 - Finding Taylor Polynomials Using Technology In...Ch. 9.7 - Numerical and Graphical Approximations (a) Use the...Ch. 9.7 - Numerical and Graphical Approximations (a) Use the...Ch. 9.7 - Numerical and Graphical Approximations In...Ch. 9.7 - Numerical and Graphical Approximations In...Ch. 9.7 - Identifying Maclaurin Polynomials In Exercises...Ch. 9.7 - Identifying Maclaurin Polynomials In Exercises...Ch. 9.7 - Identifying Maclaurin Polynomials In Exercises...Ch. 9.7 - Identifying Maclaurin Polynomials In Exercises...Ch. 9.7 - Approximating a Function Value In Exercises 39-44,...Ch. 9.7 - Approximating a Function Value In Exercises 39-44,...Ch. 9.7 - Approximating a Function Value In Exercises 39-44,...Ch. 9.7 - Approximating a Function Value In Exercises 39-44,...Ch. 9.7 - Using Taylors Theorem In Exercises 45-50, use...Ch. 9.7 - Using Taylors Theorem In Exercises 45-50, use...Ch. 9.7 - Using Taylors Theorem In Exercises 45-50, use...Ch. 9.7 - Using Taylors Theorem In Exercises 45-50, use...Ch. 9.7 - Finding a Degree In Exercises 51-56, determine the...Ch. 9.7 - Finding a Degree In Exercises 4952, determine the...Ch. 9.7 - Finding a Degree In Exercises 51-56, determine the...Ch. 9.7 - Finding a Degree In Exercises 4952, determine the...Ch. 9.7 - Finding a Degree Using Technology In Exercises 57...Ch. 9.7 - Finding a Degree Using Technology In Exercises 57...Ch. 9.7 - Finding Values In Exercises 59-62, determine the...Ch. 9.7 - Finding Values In Exercises 59-62, determine the...Ch. 9.7 - Finding Values In Exercises 59-62, determine the...Ch. 9.7 - Finding Values In Exercises 59-62, determine the...Ch. 9.7 - Polynomial Approximation When an elementary...Ch. 9.7 - WRITING ABOUT CONCEPTS Taylor Polynomial State the...Ch. 9.7 - HOW DO YOU SEE IT? The figure shows the graphs of...Ch. 9.7 - Comparing Maclaurin Polynomials (a) Compare the...Ch. 9.7 - Differentiating Maclaurin Polynomials (a)...Ch. 9.7 - Graphical Reasoning The figure shows the graphs of...Ch. 9.7 - Proof Prove that if f is an odd function, then its...Ch. 9.7 - Proof Prove that if f is an even function, then...Ch. 9.7 - Proof Let Pn(x) be the nth Taylor polynomial for f...Ch. 9.7 - WritingThe proof in Exercise 70 guarantees that...Ch. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Center of a Power Series In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Radius of Convergence In Exercise 39...Ch. 9.8 - Finding the Radius of Convergence In Exercise 39...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Finding the Interval of Convergence In Exercises...Ch. 9.8 - Writing an Equivalent Series In Exercises 45-48,...Ch. 9.8 - Writing an Equivalent Series In Exercises 45-48,...Ch. 9.8 - Writing an Equivalent Series In Exercises 4144,...Ch. 9.8 - Writing an Equivalent Series In Exercises 4144,...Ch. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - Finding Intervals of Convergence In Exercises...Ch. 9.8 - WRITING ABOUT CONCEPTS Power Series Define a power...Ch. 9.8 - WRITING ABOUT CONCEPTS Radius of Convergence...Ch. 9.8 - WRITING ABOUT CONCEPTS Interval of Convergence...Ch. 9.8 - WRITING ABOUT CONCEPTS Domain of a Power Series...Ch. 9.8 - WRITING ABOUT CONCEPTS Using a Power Series...Ch. 9.8 - Conditional or Absolute ConvergenceGive examples...Ch. 9.8 - Writing a Power Series Write a power series that...Ch. 9.8 - HOW DO YOU SEE IT? Match the graph of the first 10...Ch. 9.8 - Using Power Series Let f(x)n=1(1)nx2n+1(2n+1)!...Ch. 9.8 - Using a Power SeriesLet f(x)=n=0xnn!. (a) Find the...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Differential Equation In Exercises 59-64, show...Ch. 9.8 - Bessel FunctionThe Bessel function of order 0 is...Ch. 9.8 - Bessel Function The Bessel function of order 1 is...Ch. 9.8 - Investigation The interval of convergence of the...Ch. 9.8 - Investigation The interval of convergence of the...Ch. 9.8 - Identifying a Function In Exercises 6972, the...Ch. 9.8 - Identifying a Function In Exercises 69-72, the...Ch. 9.8 - Identifying a Function In Exercises 69-72, the...Ch. 9.8 - Identifying a Function In Exercises 6972, the...Ch. 9.8 - True or False? In Exercises 73-76, determine...Ch. 9.8 - True or False? In Exercises 73-76, determine...Ch. 9.8 - True or False? In Exercises 73-76, determine...Ch. 9.8 - True or False? In Exercises 73-76, determine...Ch. 9.8 - Proof Prove that the power series...Ch. 9.8 - Using a Power Series Let g(x)=1+2x+x2+2x3+x4+...Ch. 9.8 - Using a Power Series Let f(x)=n=pcnxn where cn+1cn...Ch. 9.8 - Proof Prove that if the power series n=0cnxn has a...Ch. 9.8 - Proof For n0, let R0 and cn0. Prove that if the...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Geometric Power Series In Exercises 3-6,...Ch. 9.9 - Finding a Power Series In Exercises 516, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 516, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 516, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 7-18, find a...Ch. 9.9 - Finding a Power Series In Exercises 516, find a...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Using a Power Series In Exercises 19-28, use the...Ch. 9.9 - Graphical and Numerical Analysis In Exercises 29...Ch. 9.9 - Graphical and Numerical Analysis In Exercises 29...Ch. 9.9 - Approximating a Sum In Exercises 31 and 32, (a)...Ch. 9.9 - Approximating a Sum In Exercises 31 and 32, (a)...Ch. 9.9 - Approximating a Value In Exercises 33-36, use the...Ch. 9.9 - Approximating a Value In Exercises 33-36, use the...Ch. 9.9 - Approximating a Value In Exercises 33-36, use the...Ch. 9.9 - Approximating a Value In Exercises 33-36, use the...Ch. 9.9 - Using a Power Series In Exercises 37-40, use the...Ch. 9.9 - Using a Power Series In Exercises 37-40, use the...Ch. 9.9 - Using a Power Series In Exercises 37-40, use the...Ch. 9.9 - Using a Power Series In Exercises 37-40, use the...Ch. 9.9 - Probability A fair coin is tossed repeatedly. The...Ch. 9.9 - Finding the Sum of a Series Use die results of...Ch. 9.9 - Writing In Exercises 4144, explain how to use the...Ch. 9.9 - Writing In Exercises 4144. explain how to use the...Ch. 9.9 - Writing In Exercises 4144. explain how to use the...Ch. 9.9 - Writing In Exercises 4144. explain how to use the...Ch. 9.9 - Proof Prove that arctanx+arctany=arctanx+y1xy for...Ch. 9.9 - Verifying an Identity Use the result of Exercise...Ch. 9.9 - Approximating Pi In Exercises 45 and 46, (a) use...Ch. 9.9 - Approximating Pi In Exercises 45 and 46, (a) use...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Finding the Sum of a Series In Exerciser 47-52,...Ch. 9.9 - Using Series One of the series in Exercises 47-52...Ch. 9.9 - Radius of Convergence The radius of convergence of...Ch. 9.9 - Convergence of a Power Series The power series...Ch. 9.9 - HOW DO YOU SEE IT? 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Proof In Exercises 13-16, prove that the Maclaurin...Ch. 9.10 - Proof In Exercises 13-16, prove that the Maclaurin...Ch. 9.10 - Proof In Exercises 13-16, prove that the Maclaurin...Ch. 9.10 - Proof In Exercises 13-16, prove that the Maclaurin...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 17–26, use...Ch. 9.10 - Using a Binomial Series In Exercises 1726, use the...Ch. 9.10 - Using a Binomial Series In Exercises 1726, use the...Ch. 9.10 - Using a Binomial Series In Exercises 1726, use the...Ch. 9.10 - Using a Binomial Series In Exercises 1726, use the...Ch. 9.10 - Finding a Maclaurin Series In Exercises 27-40,...Ch. 9.10 - Finding a Maclaurin Series In Exercises 2740. find...Ch. 9.10 - 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Finding a Maclaurin Series In Exercises 41-44,...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding Terms of a Maclaurin Series In Exercises...Ch. 9.10 - Finding a Maclaurin Series In Exercises 53 and 54....Ch. 9.10 - Finding a Maclaurin Series In Exercises 53 and 54....Ch. 9.10 - Verifying a Sum In Exercises 55-58, verify the...Ch. 9.10 - Verifying a Sum In Exercises 55-58, verify the...Ch. 9.10 - Verifying a Sum In Exercises 55-58, verify the...Ch. 9.10 - Verifying a Sum In Exercises 55-58, verify the...Ch. 9.10 - Finding a Limit In Exercises 59-62, use the series...Ch. 9.10 - Finding a Limit In Exercises 59-62, use the series...Ch. 9.10 - Finding a Limit In Exercises 59-62, use the series...Ch. 9.10 - Finding a Limit In Exercises 59-62, use the series...Ch. 9.10 - 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Binomial Series Define the binomial series. 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(a) Find the power series...Ch. 9.10 - 87. Proof Prove that for any real x.
Ch. 9.10 - 88. Finding a Maclaurin Series Find the Maclaurin...Ch. 9.10 - Evaluating a Binomial Coefficient In Exercises...Ch. 9.10 - Evaluating a Binomial Coefficient In Exercises...Ch. 9.10 - Evaluating a Binomial Coefficient In Exercises...Ch. 9.10 - Evaluating a Binomial Coefficient In Exercises...Ch. 9.10 - 93. Writing a Power Series Write the power series...Ch. 9.10 - 94. Proof Prove that e is irrational. [Hint:...Ch. 9.10 - 95. Using Fibonacci Numbers Show that the...Ch. 9.10 - PUTNAM EXAM CHALLENGE
96. 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Graphical, and Analytic Analysis In...Ch. 9 - Numerical, Graphical, and Analytic Analysis In...Ch. 9 - Numerical, Graphical, and Analytic Analysis In...Ch. 9 - Numerical. Graphical, and Analytic Analysis In...Ch. 9 - Finding the Sum of a Convergent Series In...Ch. 9 - Finding the Sum of a Convergent Series In...Ch. 9 - Finding the Sum of a Convergent Series In...Ch. 9 - Finding the Sum of a Convergent Series In...Ch. 9 - Using a Geometric Series In Exercises 35 and 36,...Ch. 9 - Using a Geometric Series In Exercises 55 and 36,...Ch. 9 - Using a Geometric Series or the nth-Term Test In...Ch. 9 - Using Geometric Series or the nth-Term Test In...Ch. 9 - Using a Geometric Series or the nth-Term Test In...Ch. 9 - Using Geometric Series or the nth-Term Test In...Ch. 9 - Compound InterestA deposit of $125 is made at the...Ch. 9 - Distance A ball is dropped from a height of 8...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Integral Test or a p-Series In Exercises...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Direct Comparison Test or the Limit...Ch. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Using the Alternating Series Test In Exercises 55...Ch. 9 - Using the Alternating Series Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Using the Ratio Test or the Root Test In Exercises...Ch. 9 - Numerical. Graphical, and Analytic Analysis In...Ch. 9 - Numerical, Graphical, and Analytic Analysis In...Ch. 9 - Review In Exercises 71-76, determine the...Ch. 9 - Finding a Maclaurin Polynomial In Exercises 69 and...Ch. 9 - Finding a Taylor Polynomial In Exercises 71 and...Ch. 9 - Finding a Taylor Polynomial In Exercises 79 and...Ch. 9 - Finding a Degree In Exercises 81 and 82, determine...Ch. 9 - Finding a Degree In Exercises 81 and 82, determine...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding the Interval of Convergence In Exercises...Ch. 9 - Finding Intervals of Convergence In Exercises 89...Ch. 9 - Finding Intervals of Convergence In Exercises 89...Ch. 9 - Differential Equation In Exercises 91 and 92, show...Ch. 9 - Differential Equation In Exercises 91 and 92, show...Ch. 9 - Finding a Geometric Power Series In Exercises 93...Ch. 9 - Finding a Geometric Power Series In Exercises 93...Ch. 9 - Finding a Power Series In Exercises 95 and 96,...Ch. 9 - Finding a Power Series In Exercises 87 and 88,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding the Sum of a Series In Exercises 97-102,...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Finding a Taylor Series In Exercises 103-110, use...Ch. 9 - Forming Maclaurin Series Determine the first four...Ch. 9 - Forming Maclaurin Series Determine the first four...Ch. 9 - Finding a Maclaurin Series In Exercises 113-116,...Ch. 9 - Finding a Maclaurin Series In Exercises 113-116,...Ch. 9 - Finding a Maclaurin Series In Exercises 105108,...Ch. 9 - Finding a Maclaurin Series In Exercises 113- 116,...Ch. 9 - Finding a Limit In Exercises 109 and 110, use the...Ch. 9 - Finding a Limit In Exercises 109 and 110, use the...Ch. 9 - Cantor Set The Cantor set (Georg Cantor,...Ch. 9 - Using Sequences (a) Given that limna2n and...Ch. 9 - Using a Series It can be shown that n=11n2=26 [see...Ch. 9 - Finding a Limit Let T be an equilateral triangle...Ch. 9 - Using Center of Gravity Identical blocks of unit...Ch. 9 - Using Power Series (a) Consider the power series...Ch. 9 - Finding Sums of Series (a) Find a power series for...Ch. 9 - Using the Alternating Series Test The graph of the...Ch. 9 - Conditional and Absolute Convergence For what...Ch. 9 - Proof (a) Consider the following sequence of...Ch. 9 - Proof Let { an} be a sequence of positive numbers...Ch. 9 - Using a Series Consider the infinite series...Ch. 9 - Deriving Identities Derive each identity using the...Ch. 9 - Population Consider an idealized population with...Ch. 9 - Spheres Imagine you are stacking an infinite...Ch. 9 - Determining Convergence or Divergence (a)...

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