Math 70 Final Eam Review Remember that you are require to evaluate this class by going to evaluate.unt.eu an filling out the survey before minight May 8. It will only take between 5 an 0 minutes, epening on how many comments you wish to inclue.. Does the sequence below converge? If so fin its limit. n +3 a n = 3n. Fin the formula for the n th term of the sequence that starts, 3 5, 5 5, 7 5, 3. Does the sequence below converge? If so fin its limit. c n = n 3n + 4. Compute ( ) n 3 n/ 5. Compute 4n 6. Determine if the series converges or iverges. n 3n 3 + n + 7. Determine if the series converges or iverges. 7n + 00 8. Determine if the series converges or iverges. ( + ) n n 9. Determine if the series converges or iverges. n n 3 +5n + 0. Determine if the series converges or iverges. n 3 e n. Determine if the series converges or iverges. (n!) 3 (3n)!. For which values of p oes the series converge? n= n(ln n)(ln ln n) p 3. Determine if the series converges absolutely, converges, or iverges. Justify carefully. ( ) n ( n + n)
Math 70 Final Eam Review 4. Determine if the series converges absolutely, converges, or iverges. Justify carefully. cos n n 3/ 5. How many terms o you nee to a to get a number that is within 0.0 of the ( ) n infinite sum for n +3? 6. Fin the raius an interval of convergence: n ( +)n n + 7. Fin the raius an interval of convergence: n n n+ 8. Fin the interval of convergence an the raius of convergence for a) b) ( ) n 3 n n 5 n. ( ) n n 3 n5 n. 9. Fin the interval of convergence an the raius of convergence for ( ) n n. n! 0. Fin the interval of convergence an the raius of convergence for () n n.. Compute the erivative of the power series an compare the intervals of convergence of the series an its erivative. ( ) n n. Compute the following: (e arcsin ) 3. Compute the following: log (0 + ) 4. Compute the following: ln(sec ) 5. Compute in two ifferent ways: 6. Compute: cot() 7. Compute the following: ln 4 8. Compute the following: (4 4 )
Math 70 Final Eam Review 3 9. Compute the following: + 9 4 30. Compute the following: arcsin() 3. Compute the following: arctan 3. Fin the area uner the graph of y =ln an above the ais for e. 33. Fin the volume of the soli of revolution obtaine by rotating the region boune by the coorinate aes, y =3,an = y + about the y-aes. 34. Fin the length of the segment of the parabola y = with 0. 35. A bug ries on the perimeter of a wheel of raius along a straight roa. What is the length of the path travele by the bug every time the wheel turns one complete revolution? 36. Solve the ifferential equation: y = y 37. Solve the ifferential equation: y = y 38. Show the erivation for the Maclaurin epansion for sin epane at 0. 39. Derive the Taylor series for ln centere at = e. 40. Use algebra an known Taylor series to erive the Taylor series for cos(3 ). 4. Derive the Maclaurin series for ln( + ). 4. Fin the power series centere at 0 for arctan(). (First fin the series for its erivative using know series.) 43. Use algebra an known Taylor series to erive the Taylor series for sin(). 44. Use the Taylor series to show e = e. 45. Use the remainer theorem to estimate the error from using the egree four Maclaurin approimation of cos for 0.5 <<0.5. Can you use the alternating series error estimate to give another error estimate for this situation? If so, how o the two compare, if not, why not? 46. Use the remainer theorem to estimate the egree of the Maclaurin polynomial neee to approimate e to within 0.0000 for 0.5. 47. Show that for an even function, all the coefficients with an o subscript in the Maclaurin epansion of the function are 0. What is the corresponing statement for even functions? 48. Fin the first four terms of the binomial series for ( ) 4/5.
Math 70 Final Eam Review 4 49. Fin the first four terms of the binomial series for ( + ) 3/. 50. Fin the first four terms of the binomial series for ( ) 3/5. 5. Use power series to solve the initial value problem y + y =, y(0) =. 5. Compute the following: log 0 53. Compute the following: t 4 + t t t 3 + t 54. Compute the following: 5 sin(3) 55. Compute the following: sin( t )e3t t 56. Compute the following: 3 ln 57. Compute the following: ( 3+) ( +) 58. Compute the following: e 59. Compute the following: sin(ln )) 60. Compute the following: 4 + 3 6. Compute the following: (5 ) 6. Compute the following: 3 + 63. Compute the following: 4 + + 64. Compute the following: sin +cos 65. Compute the following: 4 66. Compute the following: +6 +0
Math 70 Final Eam Review 5 67. Compute the following: ln( + ) 68. Compute the following: 3 4 3 0 +3 5 3 0 69. Compute the following: sec 6 70. Compute the following: tan 5 77. Compute: lim /( ) 78. Compute: lim et t 79. For which values of p oes the integral 0 ln p converge? 80. Compute: lim (log ( +) ) 8. Use Maclaurin series an properties of series to prove sin(a + ) = sin(a)cos()+cos(a)sin(). 8. How many roots oes the function f() =e ( +4) + 0 have? Prove your answer. 7. Compute the following: sin cos 5 7. Compute the following: sin cos 4 73. Compute 74. Compute lim π lim ( +0 ) cos π/ 75. Compute lim 0 +( 3) 5 76. Compute: lim 0 ln( + )