S OLUTIONS M ANUAL M ULTIVARIABLE E LKA B LOCK F RANK P URCELL T HOMAS ’ C ALCULUS T HIRTEENTH E DITION AND T HOMAS ’ C ALCULUS E ARLY T RANSCENDENTALS T HIRTEENTH E DITION Based on the original work by George B. Thomas, Jr Massachusetts Institute of Technology as revised by Maurice D. Weir Naval Postgraduate School Joel Hass University of California, Davis with the assistance of Christopher Heil Georgia Institute of Technology Learning Objectives L-1 LEARNING OBJECTIVES CHAPTER 10. Infinite Sequences and Series Section 1. Sequences 1. Find terms of a sequence. 2. Find the formula for the nth term of a sequence. 3. Determine if a sequence is monotonic and bounded. 4. Determine if a sequence converges or diverges. 5. Find the limit of a sequence, if one exists. 6. Find the limit of a recursively defined sequence. 7. Solve theory and application problems involving sequences. Section 2. Infinite Series 1. Find the formula for the nth partial sum of a series. 2. Find the sum of a series, if it converges. 3. Express repeating decimals as the ratio of two integers. 4. Use the n th-term test for divergence. 5. Find the sum of a geometric series and the values for which it converges. 6. Solve theory and application problems involving series. Section 3. The Integral Test 1. Use the integral test to determine if a series converges or diverges. 2. Estimate bounds for the remainder when using the integral test. 3. Use the integral test to solve theory and application problems involving series. Section 4. Comparison Tests 1. Use the comparison test to determine if a series converges or diverges. 2. Use the limit comparison test to determine if a series converges or diverges. 3. Use cmparison tests to solve theory and application problems involving series. Section 5. Absolute Convergence; The Ratio and Root Tests 1. Use the Ratio Test to determine whether a series converges absolutely or diverges. 2. Use the Root Test to determine whether a series converges absolutely or diverges. 3. Solve theory problems involving the Root and Ratio Tests. Section 6. Alternating Series, Absolute and Conditional Convergence 1. Determine if a series converges absolutely, converges conditionally, or diverges. 2. Estimate the error in approximating the sum of an alternating series. 3. Determine the number of terms needed to estimate the sum of an alternating series. 4. Approximate the sum of an alternating series given a specific magnitude of error. 5. Solve theory and application problems involving alternating series. Section 7. Power Series 1. Find the radius and interval of convergence of a power series. 2. Determine whether a power series diverges, converges conditionally, or converges absolutely at the endpoints of the interval of convergence. L-2 Learning Objectives 3. Use algebraic operations, term-by-term differentiation, and term-by-term integration to find the sum of a power series. 4. Solve theory and application problems involving power series. Section 8. Taylor and Maclaurin Series 1. Find the n th Taylor polynomial for a function f at a point x = a . 2. Find the Taylor series for a function f at a point x = a . 3. Find the Maclaurin series for a function f . 4. Find the values of x for which a Taylor or Maclaurin series converges absolutely. 5. Solve theory problems involving Taylor or Maclaurin series. Section 9. Convergence of Taylor Series 1. Use substitution and power series operations to find a Taylor series. 2. Show that a Taylor series converges at a given point by estimating the remainder term. 3. Estimate the error when f ( x ) is approximated by the n th Taylor polynomial P n ( x ). 4. Determine how large n must be in order that the Taylor polynomial P n ( x ) approximate f ( x ) to within a given accuracy. 5. Solve theory and application problems involving Taylor series. Section 10. The Binomial Series and Applications of Taylor Series 1. Find terms of a binomial series. 2. Find a binomial series. 3. Use series to estimate the value of an integral within a specific error. 4. Find a polynomial that will approximate a function given by an integral to a given accuracy. 5. Use series to evaluate limits that involve indeterminate forms. 6. Use algebraic operations and common Taylor series to find the sum of a given series. 7. Solve theory and application problems involving Taylor series. 8. Use Euler's identity. CHAPTER 11. Parametric Equations and Polar Coordinates Section 1. Parametrizations of Plane Curves 1. Graph a curve given by a parametric equation. 2. Find and graph a Cartesian equation corresponding to a given parametric equation. 3. Find parametric equations that define a curve or the motion of a particle. 4. Graph parametric curves using a software package. Section 2. Calculus with Parametric Curves 1. Given a parametric equation, find the parametric formulas for dy / dx and d 2 y / dx 2 . 2. Find the tangent to a curve given by a parametric equation. 3. Find the area enclosed by a parametrically defined curve. 4. Find the length of a parametrically defined curve. 5. Find the area of a surface of revolution corresponding to a parametrized curve. 6. Find the coordinates of the centroid of a region defined by a parametrized curve. 7. Solve theory and application problems involving parametric curves. Section 3. Polar Coordinates 1. Find all of the polar coordinates of a given point. Learning Objectives L-3 2. Write Cartesian coordinates for given polar coordinates. 3. Write polar coordinates for given Cartesian coordinates. 4. Graph sets of points whose polar coordinates satisfy a given equation or inequality. 5. Convert polar equations to Cartesian equations. 6. Convert Cartesian equations to polar equations. Section 4. Graphing in Polar Coordinates 1. Identify the symmetries of a curve and sketch its graph. 2. Find the slope of a curve given in polar coordinates at a given point. 3. Graph curves given in polar coordinates. Section 5. Areas and Lengths in Polar Coordinates 1. Find the area of a region enclosed by a curve given in polar coordinates. 2. Find the length of a curve given in polar coordinates. Section 6. Conic Sections 1. Sketch conic section and find quantities related to the conic section, such as vertices, foci, directrix, or asymptotes. 2. Find the standard form of a conic equation. 3. Solve problems involving shifted conic sections. 4. Solve theory and application problems related to conic sections. Section 7. Conics in Polar Coordinates 1. Find the eccentricity, foci, and directrix of a conic section. 2. Find a standard-form equation in Cartesian coordinates. 3. Find the polar equation for a conic section. 4. Graph a conic section. CHAPTER 12. Vectors and the Geometry of Space Section 1. Three-Dimensional Coordinate Systems 1. Describe the set whose coordinates satisfy the given information. 2. Find the distance between points. 3. Find the center and radius of a sphere. 4. Write an equation for a sphere. 5. Solve theory and application problems related to points in space. Section 2. Vectors 1. Find the component form of a vector. 2. Sketch vectors. 3. Find sums and scalar multiples of vectors. 4. Find the length and direction of a vector. 5. Find the midpoint of a line segment. 6. Solve theory and application problems involving vectors. Section 3. The Dot Product 1. Find the dot product of two vectors. 2. Find the angle between two vectors. L-4 Learning Objectives 3. Determine if vectors are orthogonal. 4. Find the projection of one vector onto another. 5. Solve theory and application problems involving dot products and orthogonal vectors. Section 4. The Cross Product 1. Calculate the cross product of two vectors in R 3 . 2. Find the length and direction of a cross product of two vectors. 3. Find the area of a triangle or parallelogram in space. 4. Compute a triple scalar product of three vectors. 5. Find the volume of a parallelepiped. 6. Solve theory and application problems related to cross products. Section 5. Lines and Planes in Space 1. Find parametrizations for lines and line segments in space. 2. Find the equation of a plane. 3. Find the distance from a point to a line or a plane. 4. Find the line of intersection of two planes and the angle between them. 5. Find the point at which a line meets a plane. 6. Solve theory and application problems involving lines and planes. Section 6. Cylinders and Quadric Surfaces 1. Sketch cylinders and quadric surfaces. 2. Solve theory and application problems related to cylinders and quadric surfaces. CHAPTER 13. Vector-Valued Functions and Motion in Space Section 1. Curves in Space and Their Tangents 1. Find a particle's velocity and acceleration vectors. 2. Find the angle between the velocity and acceleration vectors. 3. Find parametric equations for the line tangent to a curve. 4. Solve theory and application problems involving motion along a curve. Section 2. Integrals of Vector Functions; Projectile Motion 1. Integrate vector-valued functions. 2. Solve initial value problems. 3. Solve applications involing projectile motion. 4. Solve theory problems related to integration of vector functions. Section 3. Arc Length in Space 1. Find the arc length of a curve. 2. Find the unit tangent vector to a curve. 3. Solve theory and application problems involving arc length. Section 4. Curvature and Normal Vectors of a Curve 1. Find the unit tangent vector T , the curvature kappa, and the principal unit norm vector N for a plane curve. 2. Find the unit tangent vector T , the curvature kappa, and the principal unit norm vector N for a space curve. Learning Objectives L-5 3. Solve theory problems involving curvature. Section 5. Tangential and Normal Components of Acceleration 1. Find tangential and normal components of acceleration. 2. Find the torsion function of a smooth curve. 3. Find the TNB frame for a curve. 4. Solve theory and application problems involving acceleration. Section 6. Velocity and Acceleration in Polar Coordinates 1. Find velocity and acceleration in polar coordinates. 2. Solve problems related to Kepler's Laws. CHAPTER 14. Partial Derivatives Section 1. Functions of Several Variables 1. Evaluate a function of several variables at specified points. 2. Find the domain and range a function of two variables. 3. Sketch level curves of a function of two variables, or match level curves with a surface. 4. Sketch functions of two variables. 5. Sketch level surfaces for a function of three variables. 6. Find an equation for a level curve or level surface that passes through a given point. Section 2. Limits and Continuity in Higher Dimensions 1. Determine if the limit of a function of several variables exists, and find the limit if it does exist. 2. Determine points of continuity for functions of several variables. 3. Use the two-path test to prove the nonexistence of a limit. 4. Use the sandwich theorem to find limits. 5. Use polar coordinates to find limits. 6. Use the epsilon-delta definition of a limit. Section 3. Partial Derivatives 1. Calculate first-order partial derivatives. 2. Calculate second-order partial derivatives. 3. Use the limit definition to compute a partial derivative. 4. Use implicit differentiation to find a partial derivative. 5. Solve theory and application problems involving partial derivatives or partial differential equations. Section 4. The Chain Rule 1. Use the chain rule with one independent variable. 2. Use the chain rule with multiple independent variables. 3. Use a branch diagram to write a chain rule formula for a derivative. 4. Use implicit differentiation. 5. Find partial derivatives at specified points. 6. Apply the multi-dimensional chain rule to solve applications. L-6 Learning Objectives Section 5. Directional Derivatives and Gradient Vectors 1. Calculate the gradient of a function at a given point. 2. Find directional derivatives. 3. Find the equation for the tangent line to a level curve and illustrate with a sketch. 4. Apply knowledge of gradients and directional derivatives to solve applications. Section 6. Tangent Planes and Differentials 1. Find equations for tangent planes and normal lines to a surface. 2. Find parametric equations for the line tangent to a curve at a given point. 3. Estimate the change in a function of two or three variables. 4. Find the linearization of a function of two or three variables. 5. Find an upper bound for the error in the linearization. 6. Estimate error and sensitivity to change. 7. Solve theory and application problems related to tangent planes and differentials. Section 7. Extreme Values and Saddle Points 1. Use the first derivative test to find local extrema of a function of two variables. 2. Use the second derivative test to find local extrema and saddle points of functions of two variables. 3. Find absolute extrema of a function of two variables. 4. Find extreme values on parameterized curves. 5. Solve theory and application problems involving extreme values and saddle points. Section 8. Lagrange Multipliers 1. Solve applications involving two independent variables with one constraint. 2. Solve applications involving three independent variables with one constraint. 3. Solve applications involving three independent variables with two constraints. 4. Solve theoretical problem involving Lagrange multipliers. Section 9. Taylor's Formula for Two Variables 1. Find quadratic and cubic approximations to a function of two variables. Section 10. Partial Derivatives with Constrained Variables 1. Find partial derivatives of functions of constrained variables. CHAPTER 15. Multiple Integrals Section 1. Double and Iterated Integrals over Rectangles 1. Evaluate iterated integrals. 2. Evaluate double integrals over rectangles. 3. Find the volume beneath a surface. Section 2. Double Integrals over General Regions 1. Sketch the region of integration. 2. Find limits of integration that define a region, and write an iterated integral that gives the area of a region. 3. Evaluate integrals over a region. 4. Write an equivalent double integral with the order of integration reversed. Learning Objectives L-7 5. Evaluate an integral by reversing the order of integration. 6. Find the volume beneath a surface. 7. Evaluate an integral over an unbounded region. 8. Approximate an integral with a finite sum. 9. Solve theoretical and applied problems related to double integrals. Section 3. Area by Double Integration 1. Express the area of a region as a double integral and evaluate the integral. 2. Sketch the region indicated by the double integral, find the equations of the bounding curves, and evaluate the integral. 3. Find the average value of a function over a region. 4. Solve theory and application problems related to double integrals. Section 4. Double Integrals in Polar Form 1. Describe a region in polar coordinates. 2. Change a Cartesian integral to polar form and evaluate. 3. Change a polar integral into Cartesian form and evaluate. 4. Find the area of a region using a polar double integral. 5. Find the average value of a function using a polar integral. 6. Solve theory and application problems involving polar integrals. Section 5. Triple Integrals in Rectangular Coordinates 1. Evaluate triple integrals. 2. Write triple integrals in multiple orders of integration and evaluate. 3. Find volumes by using triple integrals. 4. Find the average value of a function of three variables. 5. Integrate by changing the order of integration. 6. Solve theory and application problems involving triple integrals. Section 6. Moments and Centers of Mass 1. Find the mass, first moments, center of mass, and moments of intertia for plates of constant or varying density. 2. Find the mass, first moments, center of mass, and moments of intertia for solids of constant or varying density. 3. Solve theory and application problems involving moments and centers of mass. Section 7. Triple Integrals in Cylindrical and Spherical Coordinates 1. Evaluate integrals in cylindrical or spherical coordinates. 2. Change the order of integration in cylindrical or spherical coordinates. 3. Find iterated integrals in cylindrical or spherical coordinates. 4. Find the volume of a solid using triple integrals. 5. Find the average value of a function over a solid. 6. Find the mass, center of mass, or moments of a solid. 7. Solve theory and application problems involving triple integrals. Section 8. Substitutions in Multiple Integrals 1. Calculate the Jacobian of a transformation and sketch the transformed region. L-8 Learning Objectives 2. Use transformations to evaluate double integrals. 3. Use transformations to evaluate triple integrals. 7. Solve theory and application problems involving substitutions in multiple integrals. CHAPTER 16. Integration in Vector Fields Section 1. Line Integrals 1. Graph vector equations. 2. Evaluate a line integral by finding a smooth parametrization of a curve. 3. Find masses and moments for coil springs, wires, and thin rods. Section 2. Vector Fields and Line Integrals: Work, Circulation, and Flux 1. Find the gradient field of a function. 2. Find a line integral of a vector field over a given curve. 3. Find the work done by a force field moving an object over a curve in space. 4. Find the flow or circulation around a curve in a velocity field. 5. Find the flux across a simple closed plane curve. 6. Find a vector field that has given properties. Section 3. Path Independence, Conservative Fields, and Potential Functions 1. Determine if a field is conservative. 2. Find a potential function for a given field. 3. Determine if a differential form is exact. 4. Use potential functions to evaluate line integrals. 5. Solve theory and application problems related to conservative fields. Section 4. Green's Theorem in the Plane 1. Verify that Green's theorem holds for a given field over a given region. 2. Find the counterclockwise circulation and outward flux for a given field over a given curve. 3. Find the work done by a field in moving a particle along a curve. 4. Using Green's Theorem to evaluate line integrals in a plane. 5. Calculate areas by using Green's theorem. 6. Solve applied problems by using Green's theorem. Section 5. Surfaces and Area 1. Find a parametrization of a surface. 2. Find the area of a surface. 3. Find a tangent plane to a parametrized surface. 4. Solve applied problems related to surfaces and area. Section 6. Surface Integrals 1. Find the surface integral of a scalar function over a given surface. 2. Find the surface integral of a vector field over a given surface. 3. Find the flux of a vector field across a given surface. 4. Find masses and moments of thin shells. Section 7. Stokes' Theorem 1. Find the curl of a vector field. Learning Objectives L-9 2. Use Stokes' theorem to find a circulation of a field around a given curve. 3. Find the integral of a curl vector field. 4. Use Stokes' theorem to calculate the flux of the curl of a field across a given surface. 5. Solve applied problems by using Stokes' theorem. Section 8. The Divergence Theorem and a Unified Theory 1. Find the divergence of a field. 2. Use the divergence theorem to calculate outward flux across the boundary of a given region. 3. Solve theory and aplication problems related to divergence. CHAPTER 17. Second-Order Differential Equations Section 1. Second-Order Linear Equations 1. Find the general solution of a second-order linear differential equation. 2. Solve initial value problems involving second-order linear differential equations. Section 2. Nonhomogeneous Linear Equations 1. Solve differential equations by the method of undetermined coefficients. 2. Solve differential equations by the method of variation of parameters. 3. Solve initial value problems by using the methods of this section. Section 3. Applications 1. Solve applications involving differential equations. Section 4. Euler Equations 1. Find the general solution to an Euler equation. 2. Solve initial value problems related to Euler equations. Section 5. Power-Series Solutions 1. Use power series to find the general solution of a differential equation. TABLE OF CONTENTS 10 Infinite Sequences and Series 701 10.1 Sequences 701 10.2 Infinite Series 712 10.3 The Integral Test 720 10.4 Comparison Tests 728 10.5 Absolute Convergence; The Ratio and Root Tests 738 10.6 Alternating Series and Conditional Convergence 744 10.7 Power Series 752 10.8 Taylor and Maclaurin Series 764 10.9 Convergence of Taylor Series 769 10.10 The Binomial Series and Applications of Taylor Series 777 Practice Exercises 786 Additional and Advanced Exercises 795 11 Parametric Equations and Polar Coordinates 801 11.1 Parametrizations of Plane Curves 801 11.2 Calculus with Parametric Curves 809 11.3 Polar Coordinates 819 11.4 Graphing Polar Coordinate Equations 825 11.5 Areas and Lengths in Polar Coordinates 832 11.6 Conic Sections 838 11.7 Conics in Polar Coordinates 849 Practice Exercises 860 Additional and Advanced Exercises 871 12 Vectors and the Geometry of Space 877 12.1 Three-Dimensional Coordinate Systems 877 12.2 Vectors 881 12.3 The Dot Product 886 12.4 The Cross Product 892 12.5 Lines and Planes in Space 898 12.6 Cylinders and Quadric Surfaces 906 Practice Exercises 912 Additional and Advanced Exercises 920 13 Vector-Valued Functions and Motion in Space 927 13.1 Curves in Space and Their Tangents 927 13.2 Integrals of Vector Functions; Projectile Motion 933 13.3 Arc Length in Space 941 13.4 Curvature and Normal Vectors of a Curve 945 13.5 Tangential and Normal Components of Acceleration 952 13.6 Velocity and Acceleration in Polar Coordinates 959 Practice Exercises 962 Additional and Advanced Exercises 969 14 Partial Derivatives 973 14.1 Functions of Several Variables 973 14.2 Limits and Continuity in Higher Dimensions 984 14.3 Partial Derivatives 990 14.4 The Chain Rule 999 14.5 Directional Derivatives and Gradient Vectors 1008 14.6 Tangent Planes and Differentials 1014 14.7 Extreme Values and Saddle Points 1024 14.8 Lagrange Multipliers 1040 14.9 Taylor's Formula for Two Variables 1052 14.10 Partial Derivatives with Constrained Variables 1055 Practice Exercises 1059 Additional and Advanced Exercises 1076 15 Multiple Integrals 1083 15.1 Double and Iterated Integrals over Rectangles 1083 15.2 Double Integrals over General Regions 1086 15.3 Area by Double Integration 1100 15.4 Double Integrals in Polar Form 1105 15.5 Triple Integrals in Rectangular Coordinates 1112 15.6 Moments and Centers of Mass 1118 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 1124 15.8 Substitutions in Multiple Integrals 1134 Practice Exercises 1142 Additional and Advanced Exercises 1149 16 Integrals and Vector Fields 1155 16.1 Line Integrals 1155 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1161 16.3 Path Independence, Conservative Fields, and Potential Functions 1172 16.4 Green's Theorem in the Plane 1178 16.5 Surfaces and Area 1185 16.6 Surface Integrals 1196 16.7 Stokes' Theorem 1206 16.8 The Divergence Theorem and a Unified Theory 1213 Practice Exercises 1219 Additional and Advanced Exercises 1230 701 CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 SEQUENCES 1. 2 1 1 1 1 0, a    2 1 2 1 2 4 2 , a     2 1 3 2 3 9 3 , a     2 3 1 4 4 16 4 a     2. 1 1 1! 1, a   1 1 2 2! 2 , a   1 1 3 3! 6 , a   1 1 4 4! 24 a   3. 2 ( 1) 1 2 1 1, a     3 ( 1) 1 2 4 1 3 , a      4 ( 1) 1 3 6 1 5 , a     5 ( 1) 1 4 8 1 7 a      4. 1 1 2 ( 1) 1, a     2 2 2 ( 1) 3, a     3 3 2 ( 1) 1, a     4 4 2 ( 1) 3 a     5. 2 2 1 1 2 2 , a   2 3 2 1 2 2 2 , a   3 4 2 1 3 2 2 , a   4 5 2 1 4 2 2 a   6. 2 1 1 1 2 2 , a    2 2 3 2 1 2 4 2 , a    3 3 7 2 1 3 8 2 , a    4 4 15 2 1 4 16 2 a    7. 1 1, a  3 1 2 2 2 1 , a    2 3 7 1 3 2 4 2 , a    3 7 15 1 4 4 8 2 , a    4 15 31 1 5 8 16 2 , a    63 6 32 , a  127 7 64 , a  255 8 128 , a  511 9 256 , a  1023 10 512 a  8. 1 1, a  1 2 2 , a    1 2 1 3 3 6 , a     1 6 1 4 4 24 , a     1 24 1 5 5 120 , a   1 6 720 , a  1 7 5040 , a  1 8 40,320 , a  1 9 362,880 , a  1 10 3,628,800 a  9. 1 2, a  2 ( 1) (2) 2 2 1, a    3 ( 1) (1) 1 3 2 2 , a       4 1 2 ( 1) 1 4 2 4 , a        5 1 4 ( 1) 1 5 2 8 , a     1 6 16 , a  1 7 32 , a   1 8 64 , a   1 9 128 , a  1 10 256 a  10. 1 2, a   1 ( 2) 2 2 1, a      2 ( 1) 2 3 3 3 , a        2 3 3 1 4 4 2 , a        1 2 4 2 5 5 5 , a      1 6 3 , a   2 7 7 , a   1 8 4 , a   2 9 9 , a   1 10 5 a   11. 1 1, a  2 1, a  3 1 1 2, a    4 2 1 3, a    5 3 2 5, a    6 8, a  7 13, a  8 21, a  9 34, a  10 55 a  12. 1 2, a  2 1, a   1 3 2 , a     1 2 1 4 1 2 , a         1 2 1 2 5 1, a     6 2, a   7 2, a  8 1, a   1 9 2 , a   1 10 2 a  13. 1 ( 1) , n n a    1, 2, n   14. ( 1) , n n a   1, 2, n   15. 1 2 ( 1) , n n a n    1, 2, n   16. 1 2 ( 1) , n n n a    1, 2, n   702 Chapter 10 Infinite Sequences and Series 17. 1 2 3( 2) , n n n a    1, 2, n   18. 2 5 ( 1) , n n n n a    1, 2, n   19. 2 1, n a n   1, 2, n   20. 4, n a n   1, 2, n   21. 4 3, n a n   1, 2, n   22. 4 2, n a n   1, 2, n   23. 3 2 ! , n n n a   1, 2, n   24. 3 1 5 , n n n a   1, 2, n   25. 1 1 ( 1) 2 , n n a     1, 2, n   26.   1 1 2 2 ( 1) 2 2 , n n n n a          1, 2, n   27.   lim 2 (0.1) 2 n n     converges (Theorem 5, #4) 28. ( 1) ( 1) lim lim 1 1 n n n n n n n                converges 29.     1 1 2 1 2 2 1 2 2 2 lim lim lim 1 n n n n n n n              converges 30.     1 1 2 2 1 1 3 3 lim lim n n n n n n n           diverges 31.   1 4 4 4 3 8 5 1 5 8 1 lim lim 5 n n n n n n n                 converges 32. 2 3 3 1 ( 3)( 2) 2 5 6 lim lim lim 0 n n n n n n n n n n               converges 33. 2 ( 1)( 1) 2 1 1 1 lim lim lim ( 1) n n n n n n n n n n                diverges 34. 1 3 2 2 70 2 1 70 4 4 lim lim n n n n n n n                       diverges 35.   lim 1 ( 1) n n    does not exist  diverges 36.   1 lim ( 1) 1 n n n    does not exist  diverges 37.         1 1 1 1 1 1 2 2 2 2 lim 1 lim 1 n n n n n n n          converges 38.     1 1 2 2 lim 2 3 6 n n n      converges 39. 1 ( 1) 2 1 lim 0 n n n       converges Section 10.1 Sequences 703 40.   ( 1) 1 2 2 lim lim 0 n n n n n        converges 41. 1 2 2 2 1 1 1 lim lim lim 2 n n n n n n n n                 converges 42.   10 1 9 (0.9) lim lim n n n n       diverges 43.     1 1 2 2 2 lim sin sin lim sin 1 n n n n                  converges 44. lim cos ( ) lim ( )( 1) n n n n n n        does not exist  diverges 45. sin lim 0 n n n   because sin 1 1 n n n n     converges by the Sandwich Theorem for sequences 46. 2 sin 2 lim 0 n n n   because 2 sin 1 2 2 0 n n n    converges by the Sandwich Theorem for sequences 47. 1 2 2 ln 2 lim lim 0 n n n n n      converges ˆ (using l'Hopital's rule) 48. 2 3 3 2 3 ln 3 3 (ln 3) 3 (ln 3) 3 6 6 3 lim lim lim lim n n n n n n n n n n n           diverges ˆ (using l'Hopital's rule) 49.         2 1 1 1 1 2 ln ( 1) 2 1 1 lim lim lim lim 0 n n n n n n n n n n n n              converges 50.     1 2 2 ln ln 2 lim lim 1 n n n n n n      converges 51. 1 lim 8 1 n n    converges (Theorem 5, #3) 52. 1 lim (0.03) 1 n n    converges (Theorem 5, #3) 53.   7 7 lim 1 n n n e     converges (Theorem 5, #5) 54.   ( 1) 1 1 lim 1 lim 1 n n n n n n e                converges (Theorem 5, #5) 55. 1 1 lim 10 lim 10 1 1 1 n n n n n n n         converges (Theorem 5, #3 and #2)