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All Textbook Solutions for Calculus: Early Transcendentals
(a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?Find the radius of convergence and interval of convergence of the series. 3.n=1(1)nnxnFind the radius of convergence and interval of convergence of the series. 4. n=1(1)nxnn3Find the radius of convergence and interval of convergence of the series. 5.n=1xn2n1Find the radius of convergence and interval of convergence of the series. 6.n=1(1)nxnn2Find the radius of convergence and interval of convergence of the series. 7.n=1xnn!Find the radius of convergence and interval of convergence of the series. 8.n=1nnxnFind the radius of convergence and interval of convergence of the series. 9.n=1xnn44nFind the radius of convergence and interval of convergence of the series. 10.n=12nn2xnFind the radius of convergence and interval of convergence of the series. 11.n=1(1)n4nnxnFind the radius of convergence and interval of convergence of the series. 12.n=1(1)n1n5nxnFind the radius of convergence and interval of convergence of the series. 13.n=1n2n(n2+1)xnFind the radius of convergence and interval of convergence of the series. 14.n=1x2nn!Find the radius of convergence and interval of convergence of the series. 15.n=0(x2)nn2+1Find the radius of convergence and interval of convergence of the series. 16.n=1(1)n(2n1)2n(x1)nFind the radius of convergence and interval of convergence of the series. 17.n=2(x+2)n2nlnnFind the radius of convergence and interval of convergence of the series. 18.n=1n8n(x+6)nFind the radius of convergence and interval of convergence of the series. 19.n=1(x2)nnnFind the radius of convergence and interval of convergence of the series. 20.n=1(2x1)n5nnFind the radius of convergence and interval of convergence of the series. 21.n=1nbn(xa)n,b0Find the radius of convergence and interval of convergence of the series. 22.n=2bnlnn(xa)n,b0Find the radius of convergence and interval of convergence of the series. 23. n=1n!(2x1)nFind the radius of convergence and interval of convergence of the series. 24. n=1n2xn246(2n)Find the radius of convergence and interval of convergence of the series. 25. n=1(5x4)nn3Find the radius of convergence and interval of convergence of the series. 26. n=2x2nn(lnn)2Find the radius of convergence and interval of convergence of the series. 27. n=1xn135(2n1)Find the radius of convergence and interval of convergence of the series. 28. n=1n!xn135(2n1)If n=0cn4n is convergent, can we conclude that each of the following series is convergent? (a) n=0cn(2)n (b) n=0cn(4)nGraph the first several partial sums sn(x) of the series n=0xn,together with the sum function f(x) = 1/(1 x), on a common screen. On what interval do these partial sums appear to be converging to f(x)?A function f is defined by f(x)=1+2x+x2+2x3+x4+ that is. its coefficients are c2n. = 1 and c2n+1 = 2 for all n 0. Find the interval of convergence of the series and find an explicit formula for f(x).If f(x)=n=0cnxn, where cn+4 = cn for all n 0, find the interval of convergence of the series and a formula for f(x).Suppose that the power series cn(xa)n satisfies cn0 for all n. Show that if limncn/cn+1 exists, then it is equal to the radius of convergence of the power series.Suppose the series cnxn has radius of convergence 2 and the series dnxn has radius of convergence 3. What is the radius of convergence of the series (cn+dn)xn?Suppose that the radius of convergence of the power series cnxn is R. What is the radius of convergence of the power series cnx2n?If the radius of convergence of the power series n=0cnxn is 10, what is the radius of convergence of the series n1ncnxn1? Why?Suppose you know that the series n=0bnxn converges for |x| 2. What can you say about the following series? Why? n=0bnn+1xn+1Find a power series representation for the function and determine the interval of convergence. f(x)=11+xFind a power series representation for the function and determine the interval of convergence. f(x)=514x2Find a power series representation for the function and determine the interval of convergence. f(x)=23xFind a power series representation for the function and determine the interval of convergence. f(x)=42x+3Find a power series representation for the function and determine the interval of convergence. f(x)=x2x4+16Find a power series representation for the function and determine the interval of convergence. f(x)=x2x+1Find a power series representation for the function and determine the interval of convergence. f(x)=x1x+2Find a power series representation for the function and determine the interval of convergence. f(x)=x+ax2+a2,a0Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. f(x)=2x4x24x+3Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. f(x)=2x+3x2+3x+2(a) Use differentiation to find a power series representation for f(x)=1(1+x)2 What is the radius of convergence? (b) Use part (a) to find a power series for f(x)=1(1+x)3 (c) Use part (b) to find a power series for f(x)=x2(1+x)3(a) Use Equation I to find a power series representation for f(x) = ln(1 - x) . What is the radius of convergence? (b) Use part (a) to find a power series for .f(x) = x ln(1 - x). (c) By pulling x=12 in your result from part (a). express In 2 as the sum of an infinite series.Find a power series representation for the function and determine the radius of convergence. f(x) =ln(5- x)Find a power series representation for the function and determine the radius of convergence. f(x)=x2tan1(x3)Find a power series representation for the function and determine the radius of convergence. f(x)=x(1+4x)2Find a power series representation for the function and determine the radius of convergence. f(x)=(x2x)3Find a power series representation for the function and determine the radius of convergence. f(x)=1+x(1x)2Find a power series representation for the function and determine the radius of convergence. f(x)=x2+x(1x)3Find a power series representation for f , and graph f and several partial sums sn(x) on the same screen. What happens as n increases? f(x)=x2x2+1Find a power series representation for f , and graph f and several partial sums sn(x) on the same screen. What happens as n increases? f(x)=ln(1+x4)Find a power series representation for f , and graph f and several partial sums sn(x) on the same screen. What happens as n increases? f(x)=ln(1+x1x)Find a power series representation for f , and graph f and several partial sums sn(x) on the same screen. What happens as n increases? f(x)=tan1(2x)Evaluate the indefinite integral as a power series. What is the radius of convergence? t1t8dtEvaluate the indefinite integral as a power series. What is the radius of convergence? t1+t3dtEvaluate the indefinite integral as a power series. What is the radius of convergence? x2ln(1+x)dxEvaluate the indefinite integral as a power series. What is the radius of convergence? tan1xdxUse a power series to approximate the definite integral to six decimal places. 00.3x1+x3dxUse a power series to approximate the definite integral to six decimal places. 01/2arctan(x/2)dxUse a power series to approximate the definite integral to six decimal places.00.2xln(1+x2)dxUse a power series to approximate the definite integral to six decimal places. 00.3x21+x4dxUse the result of Example 7 to compute arctan 0.2 correct to five decimal places.Show that the function f(x)=n=0(1)nx2n(2n)! is a solution of the differential equation fn(x)+f(x)=0(a) Show that J0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation x2J0(x)+xJ0(x)+x2J0(x)=0 (b) Evaluate01J0(x)dx correct to three decimal places.The Bessel function of order l is defined by J1(x)=n=0(1)nx2n+1n!(n+1)!22n+1 (a) Show that J1 satisfies the differential equation x2J1(x)+xJ1(x)+(x21)J1(x)=0 (b) Show that J0(x)=J1(x)(a) Show that the function f(x)=n=0xnn! is a solution of the differential equation f(x)=f(x) (b) Show that f(x) = ex.Let fn(x)=(sinnx)/n2.Show that the series fn(x) converges for all values of x but the series of derivatives fn(x) diverges when x = 2n, n an integer. For what values of x does the series fn(x) converage?Let f(x)=n=1xnn2 Find the intervals of convergence for f, f, and f.(a) Starting with lhe geometric series n=0xn, find the sum of series n=1nxn1|x|1 (b) Find the sum of each of the following series. (i)n=1nxn,|x|1 (ii)n=1n2n (c) Find the sum of each of the following series. (i) n=2n(n1)xn,|x|1 (ii) n=2n2n2n (iii) n=1n22nUse the power series for tan-1x to prove the following expression for as the sum of an infinite series: =23n=0(1)n(2n+1)3n(a) By completing the square, show that 012dxx2x+1=33 (b) By factoring x3+1 as a sum of cubes, rewrite the integral in part (a). Then express 1/(x3+1) as the sum of a power series and use it to prove the following formula for : =334n=0(1)n8(23n+1+13n+2)If f(x)=n=0bn(x5)n for all x, write a formula for b8.The graph of f is shown. (a) Explain why the series 1.60.8(x1)+0.4(x1)20.1(x1)3+ is not the Taylor series of f centered at 1. (b) Explain why the series 2.8+0.5(x2)+1.5(x2)20.1(x2)3+ is not the Taylor series of f centered at 2.If f(n)(0) = (n + 1)! for n = 0, 1, 2, , find the Maclaurin series for f and its radius of convergence.Find the Taylor series for f centered at 4 if f(n)(4)=(1)nn!3n(n+1) What is the radius of convergence of the Taylor series?Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 5. f(x) = xex, a = 0Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 6. f(x)=11+x,a=2Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 7. f(x)=x3,a=8Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 8. f(x) = ln x, a = 1Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 9. f(x) = sin x, a = /6Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. 10. f(x) =cos2 x, a = 0Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 11. f(x) = (1 x)2Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 12. f(x) = ln(1 + x)Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 13. f(x) = cos xFind the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 14. f(x) = e2xFind the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 15. f(x) = 2xFind the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 16. f(x) = x cos xFind the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 17. f(x) = sinh xFind the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 18. f(x) = cosh xFind the Taylor series for .f(x) centered at the given value of n. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 19. f(x) = x5 + 2x3 + x, a = 2Find the Taylor series for .f(x) centered at the given value of n. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 20. f(x) = x6 x4 + 2, a = 2Find the Taylor series for .f(x) centered at the given value of n. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 21. f(x) = ln x, a = 2Find the Taylor series for .f(x) centered at the given value of n. [Assume that f has a power series expansion. Do not show that Rn(x) 0.] Also find the associated radius of convergence. 22. f(x) = 1/x, a = 3Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn (x) 0.] Also find the associated radius of convergence. 23. f(x) = e2x, a = 3Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn (x) 0.] Also find the associated radius of convergence. 24. f(x) = cos x, a = /2Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn (x) 0.] Also find the associated radius of convergence. 25. f(x)=sin x, a =Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn (x) 0.] Also find the associated radius of convergence. 26. f(x)=x,a=1627EProve that the series obtained in Exercise 25 represents sin x for all x.Prove that the series obtained in Exercise 17 represents sinh x for all x.Prove that the series obtained in Exercise 18 represents cosh x for all x.Use the binomial series to expand the function as a power series. State the radius of convergence. 31. 1x4Use the binomial series to expand the function as a power series. State the radius of convergence. 32. 8+x3Use the binomial series to expand the function as a power series. State the radius of convergence. 33. 1(2+x)3Use the binomial series to expand the function as a power series. State the radius of convergence. 34. (1 x)3/4Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 35. f(x) = arctan(x2)Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 36. f(x) = sin(x/4)Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 37. f(x) = x cos 2xUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 38. f(x) = e3x e2xUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 39. f(x)=xcos(12x2)Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 40. f(x) = x2 1n(1+x3)Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 41. f(x)=x4+x2Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 42. f(x)=x22+xUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 43. f(x)=sin2x [Hint: Use sin2x=12(1cos2x).]Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. 44. f(x)={xsinxx3ifx016ifx=0Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? 45. f(x) = cos(x2)46EFind the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? 47. f(x) = xexFind the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? 48. f(x) = tan1(x3)Use the Maclaurin series for cos x to compute cos 5 correct to five decimal places.Use the Maclaurin series for ex to calculate 1/e10 correct to five decimal places(a) Use the binomial series to expand 1/1x2. (b) Use part (a) to find the Maclaurin series for sin1x.(a) Expand 1/1+x4 as a power series. (b) Use part (a) to estimate 1/1.14 correct to three decimal places.Evaluate the indefinite integral as an infinite series. 53. 1+x3dxEvaluate the indefinite integral as an infinite series. 54. x2sin(x2)dxEvaluate the indefinite integral as an infinite series. 55. cosx1xdxEvaluate the indefinite integral as an infinite series. 56. arctan(x2)dxUse series to approximate the definite integral to within the indicated accuracy. 57. 01/2x3arctanxdx (four decimal places)58EUse series to approximate the definite integral to within the indicated accuracy. 59. 00.41+x4dx (|error| 5 106)Use series to approximate the definite integral to within the indicated accuracy. 60. 00.5x2ex2dx (|error| 0.001)Use series to evaluate the limit. 61. limx0xln(1+x)x2Use series to evaluate the limit. 62. limx01cosx1+xexUse series to evaluate the limit. 63. limx0sinxx+16x3x564E65E66E67E68E69E70E71EUse multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. 72. y = ex sin2 xFind the sum of the series. 73. n=0(1)nx4nn!Find the sum of the series. 74. n=0(1)n2n62n(2n)!Find the sum of the series. 75. n=1(1)n13nn5nFind the sum of the series. 76. n=03n5nn!Find the sum of the series. 77. n=0(1)n2n+142n+1(2n+1)!Find the sum of the series. 78. 11n2+(ln2)22!(ln2)33!+Find the sum of the series. 79. 3+92!+273!+814!+Find the sum of the series. 80. 1121323+15251727+Show that if p is an nth-degree polynomial, then p(x+1)=i=0np(i)(x)i!If f(x) = ( 1 + x3)30, what is f(58)(0)?83E(a) Show that the function defined by f(x)={e1/x2ifx00ifx=0 is not equal to its Maclaurin series. (b) Graph the function in part (a) and comment on its behavior near the origin.Use the followi ng steps to prove ( 17). (a) Let g(x)=n=0(nk)xn. Differentiate this series to show that g(x)=kg(x)1+x1x1 (b) Let h(x) = (1 + x)kg(x) and show that h'(x) = 0. (c) Deduce that g(x) = (1 + x)k.86E(a) Find the Taylor polynomials up to degree 5 for f(x) = sin x centered at a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = /4, /2, and . (c) Comment on how the Taylor polynomials converge to f(x).(a) Find the Taylor polynomials up to degree 3 for f(x) = tan x centered a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = /6, /4, and /3. (c) Comment on how the Taylor polynomials converge to f(x).Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 3. f(x) =ex, a = 1Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 4. f(x) = sin x, a = /6Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 5. f(x) = cos x, a = /2Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 6. f(x) = ex sin x, a = 0Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 7. f(x) = ln x, a = 18EFind the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 9. f(x) = xe2x, a = 0Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 10. f(x) = tan1 x, a = 1(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 13. f(x) = 1/x, a = l, n = 2, 0.7 x 1.3(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 14. f(x) = x1/2, a = 4, n = 2, 3.5 x 4.5(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 15. f(x) = x2/3, a = 1, n = 3, 0.8 x 1.2(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 16. f(x) = sin x, a = /6, n = 4, 0 x /3(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 17. f(x) = sec x, a = 0, n = 2, 0.2 x 0.218E(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 19. f(x) = ex, a = 0, n = 3, 0 x 0.1(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 20. f(x) = x ln x, a = 1, n = 3, 0.5 x 1.5(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 21. f(x) = x sin x, a = 0, n = 4, 1 x 122EUse the information from Exercise 5 to estimate cos 80 correct to five decimal places. 310 Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. 5. f(x) = cos x, a = /2Use the information from Exercise 16 to estimate sin 38 correct to five decimal places. 1322 (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylors Inequality to estimate the accuracy of the approximation f(x) Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 16. f(x) = sin x, a = /6, n = 4, 0 x /3Use Taylors Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.How many terms of the Maclaurin series for ln(1 + x) do you need to use to estimate ln 1.4 to within 0.001?Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 27. sinxxx36(error0.01)Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 28. cosx1x22+x424(error0.005)Use the Alternating Series Estimation Theorem or Taylors inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. Check your answer graphically. 29. arctanxxx33+x55(error0.05)Suppose you know that f(n)(4)=(1)nn!3n(n+1) and the Taylor series of f centered at 4 converges to f(x) for all x in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates f(5) with error less than 0.0002.A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?The resistivity of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (-m). The resistivity of a given metal depends on the temperature according to the equation (t) = 20 e(t20) where t is the temperature in C. There are tables that list the values of (called the temperature coefficient) and 20 (the resistivity at 20C) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for (t) by its first- or second-degree Taylor polynomial at t = 20. (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give = 0.0039/C and 20 = 1.7 108 -m. Graph the resistivity of copper and the linear and quadratic approximations for 250C t 1000C. (c) For what values of t does the linear approximation agree with the exponential expression to within one percent?An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and q and are located at a distance d from each other, then the electric field E at the point P in the figure is E=qD2q(D+d)2 By expanding this expression for E as a series in powers of d/D, show that E is approximately proportional to 1/D3 when P is far away from the dipole.(a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos in Equation 2 by its first-degree Taylor polynomial. (b) Show that if cos is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for lo1 and li1. Also, use sin .35EA uniformly charged disk has radius R and surface charge density as in the figure. The electric potential V at a point P at a distance d along the perpendicular central axis of the disk is V=2ke(d2+R2d) where ke is a constant (called Coulombs constant). Show that VkeR2dforlargedIf a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth. (a) if R is the radius of the earth and L is the length of the highway, show that the correction is C=Rsec(L/R)R (b) Use a Taylor polynomial to show that CL22R+5L424R3 (c) Compare the corrections given by the formulas in parts (a) and (b) for a highway that is 100 km long. (Take the radius of the earth to be 6370 km.)The period of a pendulum with length L that makes a maximum angle 0 with the vertical is T=4Lg0/2dx1k2sin2x where k=sin(120) and g is the acceleration due to gravity. (In Exercise 7.7.42 we approximated this integral using Simpsons Rule.) (a) Expand the integrand as a binomial series and use the result of Exercise 7.1.50 to show that T=2Lg[1+1222k2+12322242k4+123252224262k6+] If 0 is not too large, the approximation T2L/g, obtained by using only the first term in the series, is often used. A better approximation is obtained by using two terms: T2Lg(1+14k2) (b) Notice that all the terms in the series after the first one have coefficients that are at most 14. Use this fact to compare this series with a geometric series and show that 2Lg(1+14k2)T2Lg43k244k2 (c) Use the inequalities in part (b) to estimate the period of a pendulum with L = 1 meter and 0 =10. How does it compare with the estimate T2L/g? What if 0 = 42?39E(a) What is a convergent sequence? (b) What is a convergent series? (c) What does limnan= 3 mean? (d) What does n=1an=3 mean?2RCC3RCCSuppose an=3 and sn, is the nth partial sum of the series. What is limnan? What is limn sn?5RCC6RCC7RCC8RCC9RCC10RCC11RCCWrite the binomial series expansion of ( 1 + x)k. What is the radius of convergence of this series?Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If limnan= 0, then an is convergent.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The series n=1nsin1 is convergent.3RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If cn6n is convergent, then cn(2)n is convergent.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If cn6n is convergent, then cn(6)n is convergent.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If cnxn diverges when x = 6, then it diverges when x = 10.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The Ratio Test can be used to determine whether 1/n3 converges.8RQ9RQ10RQ11RQ12RQ13RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If {an} and {bn} are divergent, then {an + bn} is divergent.Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If {an} and {bn} are divergent, then {anbn} is divergent.16RQ17RQ18RQ19RQ20RQ21RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If n=1an=A and n=1bn=B, then n=1anbn=AB.Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 1. an=2+n31+2n3Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 2. an=9n+110nDetermine whether the sequence is convergent or divergent. If it is convergent, find its limit. 3. an=n31+n2Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 4. an = cos(n/2)5RE6REDetermine whether the sequence is convergent or divergent. If it is convergent, find its limit. 7. {(1 + 3/n)4n}Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 8. {(-10)n/n!}9RE10REDetermine whether the series is convergent or divergent. 11. n=1nn3+1Determine whether the series is convergent or divergent. 12. n=1n2+1n3+1Determine whether the series is convergent or divergent. 13. n=1n35nDetermine whether the series is convergent or divergent. 14. n=1(1)nn+1Determine whether the series is convergent or divergent. 15. n=21nlnnDetermine whether the series is convergent or divergent. 16. n=1ln(n3n+1)17RE18RE19RE20RE21REDetermine whether the series is convergent or divergent. 22. n=1n+1n1n23REDetermine whether the series is conditionally convergent, absolutely convergent, or divergent. 24.n=1(1)n1n3Determine whether the series is conditionally convergent, absolutely convergent, or divergent. 25. n=1(1)n(n+1)3n22n+1Determine whether the series is conditionally convergent, absolutely convergent, or divergent. 26.n=2(1)nnInnFind the sum of the series. 27. n=1(3)n123nFind the sum of the series. 28. n=11n(n+3)29RE30RE31RE32REShow that cosh x1+12x2 for all x.34RE35RE36RE37RE38RE39REFind the radius of convergence and interval of convergence of the series. 40. n=1(1)nxnn25nFind the radius of convergence and interval of convergence of the series. 41. n=1(x+2)nn4n42RE43RE44RE45RE46RE47REFind the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex , sin x, tan1x, and ln(1 + x). 48. f(x) = tan1(x2)49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE1P2P3PLet {Pn} be a sequence of points determined as in the figure. Thus |AP1|=1, |PnPn+1|=2n1, and angle APnPn+1, is a right angle. Find limnPnAPn+1. FIGURE FOR PROBLEM 45P6P