Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
36E37E38E39E40ESuppose 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 n=1ncnxn1? Why?2EFind 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)=x2x2+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,a011E12Ea 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)3a Use Equation 1 to find a power series representation for f(x)=ln(1x). What is the radius of convergence? b Use part a to find a power series for f(x)=xln(1x). c By putting x=12 in your result from part a, express ln2 as the sum of an infinite series.15E16E17E18E19E20E21EFind 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)23EFind 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)25E26E27E28EUse 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)dx32E33EShow that the function f(x)=n=0(1)nx2n(2n)! is a solution of the differential equation f(x)+f(x)=0a 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 Evaluate 01J0(x)dx correct to three decimal places.36Ea 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) converge?39Ea Starting with the geometric series n=0xn, find the sum of the 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=1n22n41E42EIf 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 values of a. 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 values of a. 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 values of a. 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 values of a. f(x)=lnx,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 values of a. f(x)=sinx,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 values of a. f(x)=cos2x,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. f(x)=(1x)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. 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. f(x)=cosxFind 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. 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. 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. f(x)=xcosxFind 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. f(x)=sinhxFind 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. f(x)=coshxFind 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. f(x)=x5+2x3+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. f(x)=x6x4+2, 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. f(x)=lnx, 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. 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. 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. f(x)=cosx, 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. f(x)=sinx, 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. f(x)=x, a=16Prove that the series obtained in Exercise 13 represents cos x for all x.Prove 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. 1x4Use the binomial series to expand the function as a power series. State the radius of convergence. 8+x3Use the binomial series to expand the function as a power series. State the radius of convergence. 1(2+x)334EUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x)=arctan(x2)Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x)=sin(x/4)37E38EUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x)=xcos(12x2)40EUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x)=x4+x242EUse a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function. f(x)=sin2x[Hint:Usesin2x=12(1cos2x).]44E45E46E47EFind 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? 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. 1+x3dxEvaluate the indefinite integral as an infinite series. x2sin(x2)dxEvaluate the indefinite integral as an infinite series. cosx1xdxEvaluate the indefinite integral as an infinite series. arctan(x2)dxUse series to approximate the definite integral to within the indicated accuracy. 01/2x3arctanxdx(fourdecimalplaces)Use series to approximate the definite integral to within the indicated accuracy. 01sin(x4)dx(fourdecimalplaces)59EUse series to approximate the definite integral to within the indicated accuracy. 00.5x2ex2dx(|error|0.001)Use series to evaluate the limit. limx0xln(1+x)x2Use series to evaluate the limit. limx01cosx1+xexUse series to evaluate the limit. limx0sinxx+16x3x564EUse series to evaluate the limit. limx0x33x+3tan1xx566EUse multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y=ex2cosx68EUse multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y=xsinxUse multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y=exln(1+x)Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y=(arctanx)272EFind the sum of the series. n=0(1)nx4nn!74E75E76E77E78EFind the sum of the series. 3+92!+273!+814!+...80E81EIf f(x)=(1+x3)30, what is f(58)(0)?83Ea 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.85E86E1Ea Find the Taylor polynomials up to degree 3 for f(x)=tanx centered at 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).3E4E5E6E7E8E9E10EUse a computer algebra system to find the Taylor polynomials Tn centered at a for n=2,3,4,5. Then graph these polynomials and f on the same screen. f(x)=cotx,a=/412E13E14E15E16E17E18Ea 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)|. f(x)=ex2,a=0,n=3,0x0.120E21E22E23E24EUse Taylors Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001.26E27E28E29ESuppose you know that f(n)(4)=(1)nn!3n(n+1) and the Taylor series of 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 2m/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)=20e(t20) where t is the temperature in C. There are tables that list the values of called the temperature coefficient and p 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=1.7108-m. Graph the resistivity of copper and the linear and quadratic approximations for 250Ct1000C. 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.34EIf a water wave with length L moves with velocity v across a body of water with depth d, as in the figure on page 822, then v2=gL2tanh2dL a If the water is deep, show that vgL/(2). b If the water is shallow, use the Maclaurin series for tanh to show that vgd. Thus in shallow water the velocity of a wave tends to be independent of the length of the wave. c Use the Alternating Series Estimation Theorem to show that if L10d, then the estimate v2gd is accurate to within 0.014gL.36EIf 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=2Lg22[1+1222k2+12322242k4+123252224262k6+] If 0 is not too large, the approximation T=2L/g, obtained by using only the first term in the series, is often used. A better approximation is obtained by using two terms: T=2Lg(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?39Ea What is a convergent sequence? b What is a convergent series? c What does limnan=3 mean? d What does n=1an=3 mean?2CC3CC4CC5CC6CC7CCa Write the general form of a power series. b What is the radius of convergence of a power series? c What is the interval of convergence of a power series?9CC10CC11CC12CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQ10TFQ11TFQDetermine 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 is divergent, then |an| 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 f(x)=2xx2+13x3 converges for all x, then f(0)=2.14TFQ15TFQ16TFQ17TFQDetermine 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 an0 and limn(an+1/an)1, then limnan=0.19TFQ20TFQ21TFQ22TFQ1E2E3E4E5E6E7E8E9E10E11E12E13E14EDetermine whether the series is convergent or divergent. n=21nlnn16EDetermine whether the series is convergent or divergent. n=1cos3n1+(1.2)n18E19E20EDetermine whether the series is convergent or divergent. n=1(1)n1nn+122E23E24E25E26E27E28E29E30EFind the sum of the series. 1e+e22!e33!+e44!32E33E34E35Ea Find the partial sum s5 of the series n=11/n6 and estimate the error in using it as an approximation to the sum of the series. b Find the sum of this series correct to five decimal places.37E38E39E40E41E42E43E44EFind the Taylor series of f(x)=sinxata=/6.46E47EFind 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,sinx,tan1x, and ln(1+x). f(x)=tan1(x2)49E50EFind 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,sinx,tan1x, and ln(1+x). f(x)=sin(x4)52E53E54E55E56E57E58E59EThe force due to gravity on an object with mass m at a height h above the surface of the earth is F=mgR2(R+h)2 where R is the radius of the earth and g is the acceleration due to gravity for an object on the surface of the earth. a Express F as a series in powers of h/R. b Observe that if we approximate F by the first term in the series, we get the expression Fmg that is usually used when h is much smaller than R. Use the Alternating Series Estimation Theorem to estimate the range of values of h for which the approximation Fmg is accurate to within one percent. Use R = 6400 km.61EIf f(x)=ex2, show that f(2n)(0)=(2n)!n!.If f(x)=sin(x3), find f(15)(0).A function f is defined by f(x)=limnx2n1x2n+1 Where is f continuous?a Show that tan12x=cot12x2cotx. b Find the sum of the series n112ntanx2nLet {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.To construct the snowflake curve, start with an equilateral triangle with sides of length 1. Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part see the figure. Step 2 is to repeat step 1 for each side of the resulting polygon. This process is repeated at each succeeding step. The snowflake curve is the curve that results from repeating this process indefinitely. a Let sn,ln, and pn represent the number of sides, the length of a side, and the total length of the nth approximating curve the curve obtained after step n of the construction, respectively. Find formulas for sn,ln, and pn. b Show that pn as n. c Sum an infinite series to find the area enclosed by the snowflake curve. Note: Parts b and c show that the snowflake curve is infinitely long but encloses only a finite area.Find the sum of the series 1+12+13+14+16+18+19+112+... where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.a Show that for xy1, arctanxarctany=arctanxy1+xy if the left side lies between /2 and /2. b Show that arctan120119arctan1239=/4. c Deduce the following formula of John Machin (16801751): 4arctan15arctan1239=4 d Use the Maclaurin series for arctan to show that 0.1973955597arctan150.1973955616 e Show that 0.004184075arctan12390.004184077 f Deduce that, correct to seven decimal places, 3.1415927. Machin used this method in 1706 to find correct to 100 decimal places. Recently, with the aid of computers, the value of has been computed to increasingly greater accuracy. In 2013 Shigeru Kondo and Alexander Yee computed the value of to more than 12 trillion decimal places8PUse the result of Problem 7a to find the sum of the series n=1arctan(2/n2).10P11PSuppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show that it is possible to do this so that the top book extends entirely beyond the table. In fact, show that the top book can extend any distance at all beyond the edge of the table if the stack is high enough. Use the following method of stacking: The top book extends half its length beyond the second book. The second book extends a quarter of its length beyond the third. The third extends one-sixth of its length beyond the fourth, and so on. Try it yourself with a deck of cards. Consider centers of mass.13P14PSuppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. The figure illustrates the case n=4. If A is the area of the triangle and An is the total area occupied by the n rows of circles, show that limnAnA=23A sequence {an} is defined recursively by the equations a0=a1=1 n(n1)an=(n1)(n2)an1(n3)an2 Find the sum of the series n=0an.17P18PFind the sum of the series n=1(1)n(2n+1)3n.20PFind all the solutions of the equation 1+x2!+x24!+x36!+x48!+...=0 Hint: Consider the cases x0 and x0 separately.22PConsider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less than 90.a Show that the Maclaurin series of the function f(x)=x1xx2 is n=1fnxn where fn, is the nth Fibonacci number, that is, f1=1, f2=1, and fn=fn1+fn2 for n3. Hint: Write x/(1xx2)=c0+c1x+c2x2+... and multiply both sides of this equation by 1xx2. b By writing f(x) as a sum of partial fractions and thereby obtaining the Maclaurin series in a different way, find an explicit formula for the nth Fibonacci number.Let u=1+x33!+x66!+x99!+... v=x+x44!+x77!+x1010!+... w=x22!+x55!+x88!+... Show that u3+v3+w33uvw=1.Prove that if n1, the nth partial sum of the harmonic series is not an integer. Hint: Let 2k be the largest power of 2 that is less than or equal to n and let M be the product of all odd integers that are less than or equal to n. Suppose that Sn=m, an integer. Then M2ksn=M2km. The right side of this equation is even. Prove that the left side is odd by showing that each of its terms is an even integer, except for the last one.Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?Sketch the points (1,5,3),(0,2,3),(3,0,2), and (2,2,1) on a single set of coordinate axes.Which of the points A(4,0,1),B(3,1,5), and C(2,4,6) is closest to the yz-plane? Which point lies in the xz-plane?What are the projections of the point 2, 3, 5 on the xy-, yz-,and xz-planes? Draw a rectangular box with the origin and 2, 3, 5 as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.