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All Textbook Solutions for Calculus (MindTap Course List)
Finding the Number of Terms In Exercises 35-40, use Theorem 9.15 to determine the number of terms required to approximate the sum of the series with an error of less than 0.001. n=1(1)n+1n5Finding the Number of Terms In Exercises 35-40, use Theorem 9.15 to determine the number of terms required to approximate the sum of the series with an error of less than 0.001. n=0(1)nn!Finding the Number of Terms In Exercises 35-40, use Theorem 9.15 to determine the number of terms required to approximate the sum of the series with an error of less than 0.001. n=0(1)n(2n)!Determining Absolute and Conditional Convergence In Exercises 41-58, determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n2nDetermining Absolute and Conditional Convergence In Exercises 41-58, determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1n2Determining Absolute and Conditional Convergence In Exercises 41-58, determine whether the series converges absolutely or conditionally, or diverges. n=1(1)nn!Determining Absolute and Conditional Convergence In Exercises 41-58, determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1n+3Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1nDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1nnDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1n2(n+1)248EDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=2(1)nnlnnDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=0(1)nenDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=2(1)nnn35Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n1n4/3Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=0(1)n(2n+1)!Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=0(1)nn+4Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=0cosxn+1Determining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1(1)n+1arctann57EDetermining Absolute end Conditional Convergence In Exercises 41-58. determine whether the series converges absolutely or conditionally, or diverges. n=1sin[(2n1)/2]n59E60E61EHOW DO YOU SEE IT? The graphs of the sequences of partial sums of two series are shown in the figures. Which graph represents the partial sums of an alternating series? Explain.63E64EProof Prove that if | an | converges, then an2 converges. Is the converse true? If not, give an example that shows it is false.66E67EFinding Values Find all values of x for which the series (xn/n) (a) converges absolutely and (b) converges conditionally.69E70E71E72E73E74E75E76E77E78E79E80E81E82ECONCEPT CHECK Ratio and Root Tests In Exercises 1-6, what can you conclude about the convergence or divergence of an?. limn|an+1an|=02E3E4E5E6E7E8E9EMatching In Exercises 9-14, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b). (c), (d), (e), and (f).] n=1(34)n(1n!)Matching In Exercises 9-14, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b). (c), (d), (e), and (f).] n=1(3)n+1n!Matching In Exercises 9-14, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b). (c), (d), (e), and (f).] n=1(1)n14(2n)!Matching In Exercises 9-14, match the series with the graph of its sequence of partial sums. [The graphs are labeled (a), (b). (c), (d), (e), and (f).] n=1(4n5n3)n14ENumerical, Graphical, and Analytic Analysis In Exercises 15 and 16, (a) use the Ratio Test to verify that the series converges, (b) use a graphing utility to find the indicated partial sum Sn and complete the table, (c) use a graphing utility to graph the first 10 terms of the sequence of partial sums, (d) use the table to estimate the sum of the series, and (e) explain the relationship between the magnitudes of the terms of the series and the rate at which the sequence of partial sums approaches the sum of the series. n 5 10 15 20 25 Sn n=1n3(12)n16E17E18E19E20E21E22E23E24E25E26E27EUsing the Ratio Test In Exercises 17-38, use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. n=1(1)n1(3/2)nn229E30E31E32E33E34E35E36EUsing the Ratio Test In Exercises 17-38, use the Ratio Test to determine the convergence or divergence of the series. If the Ratio Test is inconclusive, determine the convergence or divergence of the series using other methods. n=0(1)n+1n!135(2n+1)38E39EUsing the Root Test In Exercises 39-52, use the Root Test to determine the convergence or divergence of the series. n=11nn41E42E43E44E45EUsing the Root Test In Exercises 39-52, use the Root Test to determine the convergence or divergence of the series. n=0e3nUsing the Root Test In Exercises 39-52, use the Root Test to determine the convergence or divergence of the series. n=1n3nUsing the Root Test In Exercises 39-52, use the Root Test to determine the convergence or divergence of the series. n=1(n500)n49E50E51E52E53E54E55E56E57E58E59EReview In Exercises 53-70, determine the convergence or divergence of the series using any appropriate test from this chapter. Identify the test used. n=1103n361E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104E105E106E107E108ECONCEPT CHECK Polynomial Approximation An elementary function is approximated by a polynomial. In your own words, describe what is meant by saying that the polynomial is expanded about c or centered at c.2E3E4EMatching In Exercises 5-8, match the Taylor polynomial approximation of the function f(x)=ex2/2 with its graph. [The graphs are labeled (a), (b) (c), and (d).] g(x)=12x2+1Matching In Exercises 5-8, match the Taylor polynomial approximation of the function f(x)=ex2/2 with its graph. [The graphs are labeled (a), (b) (c), and (d).] g(x)=18x412x2+17EMatching In Exercises 5-8, match the Taylor polynomial approximation of the function f(x)=ex2/2 with its graph. [The graphs are labeled (a), (b) (c), and (d).] g(x)=e1/2[13(x1)3(x1)+1]Finding a First-Degree Polynomial Approximation In Exercises 9-12, find a first- degree polynomial function P1 whose value and slope agree with the value and slope of f at x=c . Use a graphing utility to graph f and P 1. f(x)=x4,c=4Finding a First-Degree Polynomial Approximation In Exercises 9-12, find a first- degree polynomial function P1 whose value and slope agree with the value and slope of f at x=c . Use a graphing utility to graph f and P 1. f(x)=6x3,c=811E12E13E14EConjecture Consider the function f(x)=cosx and its Maclaurin polynomials P2,P4. and P6 (sec Example 5). (a) Use a graphing utility to graph f and the indicated polynomial approximations. (b) Evaluate and compare the values of f(n)(0) and Pn(n)(0) for n=2,4, and 6. (c) Use the results in part (b) to make a conjecture about f(n)(0) and Pn(n)(0).Conjecture Consider the function f(x)=x2ex. (a) Find the Maclaurin polynomials P2,P3, and P4 for f. (b) Use a graphing utility to graph f,P2,P3 and P4 (c) Evaluate and compare the values of f(n)(0) and Pn(n)(0) for n=2,3, and 4. (d) Use the results in part (c) to make a conjecture about f(n)(0) and Pn(n)(0)17E18E19E20E21E22E23E24E25E26EFinding a Taylor Polynomial In Exercises 27-32, find the n th Taylor polynomial for the function, centered at c. f(x)=2x,n=3,c=1Finding a Taylor Polynomial In Exercises 27-32, find the n th Taylor polynomial for the function, centered at c. f(x)=1x2,n=4,c=2Finding a Taylor Polynomial In Exercises 27-32, find the n th Taylor polynomial for the function, centered at c. f(x)=x,n=2,c=430E31E32E33E34E35E36E37E38E39E40E41E42E43E44EUsing Taylors Theorem In Exercises 45-50, use Taylors Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. cos(0.3)1(0.3)22!+(0.3)44!46EUsing Taylors Theorem In Exercises 45-50, use Taylors Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. sinh(0.2)0.2+(0.2)33!+(0.2)55!48EUsing Taylors Theorem In Exercises 45-50, use Taylors Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. arctan(0.4)0.4(0.4)323Using Taylors Theorem In Exercises 45-50, use Taylors Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. arctan(0.4)0.4(0.4)3351E52E53E54E55E56E57E58EFinding Values In Exercises 59-62, determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x)=ex1+x+x22!+x33!x0Finding Values In Exercises 59-62, determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x)=sinxxx33!Finding Values In Exercises 59-62, determine the values of x for which the function can be replaced by the Taylor polynomial if the error cannot exceed 0.001. f(x)=cosxxx22!+x44!62E63E64E65E66E67EDifferentiating Maclaurin Polynomials (a) Differentiate the Maclaurin polynomial of degree 5 for f(x)=sinx and compare the result with the Maclaurin polynomial of degree 4 for g(x)=cosx. (b) Differentiate the Maclaurin polynomial of degree 6 for f(x)=cosx and compare the result with the Maclaurin polynomial of degree 5 for g(x)=sinx (c) Differentiate the Maclaurin polynomial of degree 4 for f(x)=ex. Describe the relationship between the two series,69E70E71E72E73E1E2E3E4EFinding the Center of a Power Series In Exercises 5-8, state where the power series is centered. n=0nxnFinding the Center of a Power Series In Exercises 5-8, state where the power series is centered. n=1(1)n(2n1)2nn!xnFinding the Center of a Power Series In Exercises 5-8, state where the power series is centered. n=1(x2)nn3Finding the Center of a Power Series In Exercises 5-8, state where the power series is centered. n=0(1)n(x)2n(2n)!Finding the Radius of Convergence In Exercises 9-14, find the radius of convergence of the power series. n=0(1)nxnn+1Finding the Radius of Convergence In Exercises 9-14, find the radius of convergence of the power series. n=0(3x)nFinding the Radius of Convergence In Exercise 9-14, find the radius of convergence of the power series. n=1(4x)nn2Finding the Radius of Convergence In Exercises 9-14, find the radius of convergence of the power series. n=0(1)nxn5nFinding the Radius of Convergence In Exercises 9-14, find the radius of convergence of the power series. n=0x2n(2n)!Finding the Radius of Convergence In Exercises 9-14, find the radius of convergence of the power series. n=0(2n)!x3nn!Finding the Interval of Convergence In Exercises 15-38, find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0(x4)nFinding the Interval of Convergence In Exercises 15, 38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0(2x)nFinding the Interval of Convergence In Exercises 15-38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=1(1)nxnnFinding the Interval of Convergence In Exercises 15-38, find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0(1)n+1(n+1)xnFinding the Interval of Convergence In Exercises 15, 38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0x5nn!Finding the Interval of Convergence In Exercises 15, 38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0(3x)n(2n)!Finding the Interval of Convergence In Exercises 15-38, find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0(2n)!(x3)nFinding the Interval of Convergence In Exercises 15, 38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=0(1)nxn(n+1)(n+2)Finding the Interval of Convergence In Exercises 15, 38 find the interval of convergence of the power series. (Be sure to include a check for convergence at the end points of the interval.) n=1(1)n1xn6nFinding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=0(1)nn!(x5)n3nFinding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=1(1)n+1(x4)nn9nFinding the Interval of Convergence In Exercise 15-38. find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=0(x3)n+1(n+1)4n+1Finding the Interval of Convergence In Exercise 15-38. find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=0(1)n+1(x1)n+1n+1Finding the Interval of Convergence In Exercise 15-38. find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=1(1)n+1(x2)nn2nFinding the Interval of Convergence In Exercise 15-38. find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=1(x3)n1(x2)n3n1Finding the Interval of Convergence In Exercises 1538, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) no(1)nx2x12n+1Finding the Interval of Convergence In Exercises 1538, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n1nn+1(2x)n1Finding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=0(1)nx2nn!Finding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=0x3n+1(3n+1)!Finding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=1n!xn(2n)!Finding the Interval of Convergence In Exercises 1538, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n=1234(n+1)xnn!Finding the Interval of Convergence In Exercises 1538, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n1[2+4+6...2n3+5+7...(2n+1)]x2n+1Finding the Interval of Convergence In Exercise 15-38, find the Interval of convergence of the power series. (Be sure in include a check for convergence at the endpoints of the interval.) n=1(1)n+13711(4n1)(x3)n4nFinding the Interval of Convergence In Exercises 1538, find the Interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) n1n!(x+1)n135...(2n1)Finding the Radius of Convergence In Exercises 39 and 40, find the radius of convergence of the power series, where c 0 and k is a positive integer. n=1(xc)n1en1Finding the Radius of Convergence In Exercises 39 and 40, find the radius of convergence of the power series, where c 0 and k is a positive integer. n=0(n!)kxn(kn)!Finding the Interval of Convergence In Exercises 41-44, find the interval of convergence of the power series, where c 0 and k is a positive integer. (Be sure to include a check for convergence at the endpoints of the interval.) n=0(xk)nFinding the Interval of Convergence In Exercises 41-44, find the interval of convergence of the power series, where c 0 and k is a positive integer. (Be sure to include a check for convergence at the endpoints of the interval.) n=0(1)n+1(xc)nncn43EFinding the Interval of Convergence In Exercises 4144, find the interval of convergence of the power series, where c 0 and k is a positive integer. (Be sure to include a check for convergence at the endpoints of the interval.) n=1n!(xc)n135...(2n1)Writing an Equivalent Series In Exercises 45-48, write an equivalent series with the index of summation beginning at n=1. n=0xnn!Writing an Equivalent Series In Exercises 45-48, write an equivalent series with the index of summation beginning at n=1. n=0(1)n+1(n+1)xn47E48E49EFinding Intervals of Convergence In Exercises 4952, find the intervals of convergence of (a) f(x), (b) f(x), (c) fn(x), and (d) f(x)dx. (Be sure to include a check for convergence at the endpoints of the intervals.) f(x)=n1(1)n+1(x5)nn5n51E52E53E54E55E56EUsing Power Series Let f(x)=n=0(1)nx2n+1(2n+1)!andg(x)=n0(1)nx2n(2n)!. (a) Find the intervals of convergence of f and g. (b) Show that f(x)=g(x)andg(x)=f(x). (c) Identify the functions f and g.58E59E60E61EDifferential Equation In Exercises 59-64, show that function represented by the power series is a solution of differential equation. y=n=0x2n(2n)!,yny=063E64EBessel Function The Bessel function of order 0 is J0(x)=k=0(1)kx2k22k(k!)2. (a) Show that the series converges for all x (b) Show that die series is a solution of the differential equation x2J0+xJ0+x2J0=0. (c) Use a graphing utility to graph die polynomial composed of the first four terms of J0 (d) Approximate 01J0dx, accurate to two decimal places.66E67EInvestigation The interval of convergence of the series n0(3x)nis(13,13). (a) Find the sum of the series when x=16. Use a graphing utility to graph the First six terms of the sequence of partial sums and the horizontal line representing the sum of the series. (b) Repeat part (a) for x=16 (c) Write a short paragraph comparing the rates of convergence of the partial sums with the sums of the series in parts (a) and (b). How do the plots of the partial sums differ as they converge toward the sum of the series? (d) Given any positive real number M, there exists a positive integer N such that the partial sum n0N(323)nM. Use a graphing utility to complete the table. M 10 100 1000 10,000 N69E70EIdentifying a Function In Exercises 69-72, the series represents a well-known function. Use a computer algebra system to graph the partial sum S10 and identify the function from the graph. f(x)=n=0(1)nxn,1x172E73E74E