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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
71E72E73E75E76EExplain why it is difficult to use the divergent series 1/k as a comparison series to test 1/(k + 1).2QCExplain how the Limit Comparison Test works.Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.What comparison series would you use with the Comparison Test to determine whether k=11k2+1 converges?4EWhat comparison series would you use with the Comparison Test to determine whether k=12k3k+1 converges?Determine whether k=21k1converges using the Comparison Test with the comparison series k=11k.What comparison series would you use with the Limit Comparison Test to determine whether k=1k2+k+5k3+3k+1 converges?8EComparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 27. k=11k2+4Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 28. k=1k2+k1k4+4k23Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 29. k=1k21k3+4Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 30. k=10.0001k+4Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 13. k=1kk2+3Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 14. k=115k+3Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 15. k=14k5k3Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 16. k=1sin2kk2Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 31. k=11k3/2+1Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 18. k=11k10kComparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 19. k=41+cos2kk3Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 20. k=1k2+k+26k(k2+1)Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 21. k=1(3k3+4)(7k2+1)(2k3+1)(4k31)Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 32. k=1kk3+1Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 33. k=1sin(1/k)k2Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 34. k=113k2kComparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 35. k=112kkComparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 36. k=11kk+2Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 27. k=12+(1)kk2Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 28. k=12+sinkkComparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 37. k=1k2+13k3+2Comparison tests Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. 38. k=21(klnk)2Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 31. k=120k3+kComparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 32. k=3lnkkComparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 33. k=112lnkComparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 34. k=2ln2kk4Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 35. k=114lnkComparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge. 36. k=15k+ek2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
Suppose 0 < ak < bk. If converges, then converges.
Suppose 0 < ak < bk. If diverges, then diverges.
Suppose 0 < bk < ck < ak. If converges, then and converge.
When applying the Limit Comparison Test, an appropriate comparison series for is .
Examining a series two ways Determine whether the follow series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer. 38. k=1e2ke4k+1Examining a series two ways Determine whether the follow series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer. 39. k=11k2+2k+140EChoose your test Use the test of your choice to determine whether the following series converge. 41. k=1(1+2k)kChoose your test Use the test of your choice to determine whether the following series converge. 42. k=1e2k+ke5kk2Choose your test Use the test of your choice to determine whether the following series converge. 47. k=1k2+2k+13k2+1Choose your test Use the test of your choice to determine whether the following series converge. 46. k=12kek1Choose your test Use the test of your choice to determine whether the following series converge. 49. k=31lnkChoose your test Use the test of your choice to determine whether the following series converge. 48. k=115k1Choose your test Use the test of your choice to determine whether the following series converge. 51. k=11k3k+1Choose your test Use the test of your choice to determine whether the following series converge. 50. k=315k3kChoose your test Use the test of your choice to determine whether the following series converge. 53. k=1(1k+2k)Choose your test Use the test of your choice to determine whether the following series converge. 54. k=25lnkkChoose your test Use the test of your choice to determine whether the following series converge. 57. k=1k8k11+3Choose your test Use the test of your choice to determine whether the following series converge. 60. k=21k2lnkChoose your test Use the test of your choice to determine whether the following series converge. 59. k=11k1+p,p0Choose your test Use the test of your choice to determine whether the following series converge. 54. k=1tan1kk2Choose your test Use the test of your choice to determine whether the following series converge. 61. k=1ln(k+2k+1)Choose your test Use the test of your choice to determine whether the following series converge. 62. k=1k1/kChoose your test Use the test of your choice to determine whether the following series converge. 63. k=21klnkChoose your test Use the test of your choice to determine whether the following series converge. 64. k=1sin21kChoose your test Use the test of your choice to determine whether the following series converge. 65. k=1tan1kChoose your test Use the test of your choice to determine whether the following series converge. 60. k=1122k+1k2Choose your test Use the test of your choice to determine whether the following series converge. 67. 113+135+157+62ESeries of squares Prove that if ak is a convergent series of positive terms, then the series ak2ak2 also converges.Two sine series Determine whether the following series converge. a. k=1sin1k b. k=1sin1kLimit Comparison Test proof Use the proof of case (1) of the Limit Comparison Test (Theorem 8.17) to prove cases (2) and (3).Infinite products An infinite product P = a1a2a3 ..., which is denoted k=1ak, is the limit of the sequence of partial products {a1, a1a2, a1a2a3, ...} Assume ak 0 for all k and L is a finite constant. a. Evaluate k=1(kk+1)=12233445. b. Show that if k=1lnak=L, then k=1ak=eL. c. Use the result of part (b) to evaluate k=0e1/2k=ee1/2e1/4e1/8.An early limit Working in the early 1600s, the mathematicians Wallis, Pascal, and Fermat were calculating the area of the region under the curve y = xP between x = 0 and x = 1, where p is a positive integer. Using arguments that predated the Fundamental Theorem of Calculus, they were able to prove that limn1nk=0n1(kn)p=1p+1. Use what you know about Riemann sums and integrals to verify this limit.Write out the first few terms of the sequence of partial sums for the alternating series 1 2 + 3 4 + 5 6 + . Does this series appear to converge or diverge?Explain why the value of a convergent alternating series, with terms that are nonincreasing in magnitude, is trapped between successive terms of the sequence of partial sums.3QCExplain why a convergent series of positive terms converges absolutely.Explain why the sequence of partial sums for an alternating series is not an increasing sequence.Describe how to apply the Alternating Series Test.3ESuppose an alternating series with terms that are nonincreasing in magnitude converges to a value L. Explain how to estimate the remainder that occurs when the series is terminated after n terms.Explain why the magnitude of the remainder in an alternating series (with terms that are nonincreasing in magnitude) is less than or equal to the magnitude of the first neglected term.Give an example of a convergent alternating series that fails to converge absolutely.Is it possible for a series of positive terms to converge conditionally? Explain.8EIs it possible for an alternating series to converge absolutely but not conditionally?Determine the values of p for which the series k=0(1)kkp converges conditionally.Alternating Series Test Determine whether the following series converge. 11. k=0(1)k2k+1Alternating Series Test Determine whether the following series converge. 12. k=1(1)kkAlternating Series Test Determine whether the following series converge. 13. k=1(1)kk3k+2Alternating Series Test Determine whether the following series converge. 14. k=1(1)k(1+1k)kAlternating Series Test Determine whether the following series converge. 15. k=1(1)k+1k3Alternating Series Test Determine whether the following series converge. 16. k=0(1)kk2+10Alternating Series Test Determine whether the following series converge. 17. k=1(1)k+1k2k3+1Alternating Series Test Determine whether the following series converge. 18. k=2(1)k1nkk2Alternating Series Test Determine whether the following series converge. 19. k=2(1)kk21k2+3Alternating Series Test Determine whether the following series converge. 20. k=0(15)kAlternating Series Test Determine whether the following series converge. 21. k=2(1)k(1+1k)Alternating Series Test Determine whether the following series converge. 22. k=1coskk2Alternating Series Test Determine whether the following series converge. 23. k=1(1)k+1k10+2k5+1k(k10+1)Alternating Series Test Determine whether the following series converge. 24. k=2(1)kk1n2kAlternating Series Test Determine whether the following series converge. 25. k=1(1)k+1k1/kAlternating Series Test Determine whether the following series converge. 28. k=1(1)kksin1kAlternating Series Test Determine whether the following series converge. 27. k=0(1)kk2+4Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S Sn| in using the nth partial sum Sn to estimate the value of the series S. 28. k=1(1)k+1k3+1; n = 3Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S Sn| in using the nth partial sum Sn to estimate the value of the series S. 29. k=1(1)kk4; n = 4Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S Sn| in using the nth partial sum Sn to estimate the value of the series S. 30. k=0(1)k(3k+1)4; n = 4Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S Sn| in using the nth partial sum Sn to estimate the value of the series S. 31. k=0(1)kk4+k+21; n = 5Estimating errors in partial sums For each of the following convergent alternating series, evaluate the nth partial sum for the given value of n. Then use Theorem 10.18 to find an upper bound for the error |S Sn| in using the nth partial sum Sn to estimate the value of the series S. 32. k=0(1)k(2k+1)!; n = 3Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 29. 1n2=k=1(1)k+1kRemainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 30. 1e=k=0(1)kk!Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 31. 4=k=0(1)k2k+1Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 32. 212=k=1(1)k+1k2Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 33. 74720=k=1(1)k+1k4Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 104 in magnitude. Although you do not need it, the exact value of the series is given in each case. 34. 232=k=0(1)k(2k+1)3Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 103. 39. k=1(1)kk5Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 103. 40. k=1(1)k(2k+1)3Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 103. 41. k=1(1)kkk2+1Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 103. 42.k=1(1)kkk4+143EEstimating infinite series Estimate the value of the following convergent series with an absolute error less than 103. 44. k=1(1)k+1(2k+1)!Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 45. k=1(1)kk2/3Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 46. k=0(1)k2kk2+9Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 47. k=0(1)k+1(k+1)!Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 48. k=1(13)kAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 49. k=1(34)kAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 50. k=1(1)k+1k0.99Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 49. k=1coskk3Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 50. k=1(1)kk2k6+1Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 51. k=1(1)ktan1kAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 52. k=1(1)kekAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 55. k=1(1)k+12k1Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 56. k=2(1)k3k1Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 53. k=1(1)kk2k+1Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 54. k=2(1)k1nkAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 55. k=1(1)ktan1kk3Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 60. k=1(1)kk2+13k4+3Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 61. k=2(1)kk2+1k31Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 62. k=1sink3k+4kAbsolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge. 63. k=1(1)k+1k!kk(Hint: Show that k!kk 2k2, for k 3.)Alternating Series Test Show that the series 1325+3749+=k=1(1)k+1k2k+1 diverges. Which condition of the Alternating Series Test is not satisfied?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If ak diverges, then |ak|diverges. e. If ak2converges, then ak converges. f. If ak0 and ak converges, then ak2 converges. g. If akconverges conditionally, then|ak|diverges.Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value 112+1314+=1n2. Show that by rearranging the terms (so the sign pattern is + + ), 1+1312+15+1714+=321n2.Alternating p-series Given that k=11k2=26, show that k=1(1)k+1k2=212. (Assume the result of Exercise 63.)Alternating p-series Given that k=11k4=490,show that k=1(1)k+1k4=74720.(Assume the result of Exercise 63.)A fallacy Explain the fallacy in the following argument. Let x=1+13+15+17+ and y=12+14+16+18+. It follows that 2y = x + y, which implies that x = y. On the other hand, xy=(112)0+(1314)0+(1516)0+0 is a sum of positive terms, so x y. Therefore, we have shown that x = y and x y.Conditions of the Alternating Series Test It can be shown that if the sequence {a2n} = {a2, a4, a6, } and the sequence {a2n 1} = {a1, a3, a5, } both converge to L, then the sequence {an} = {a1, a2, a3, } converges to L. It is also the case that if {a2n} or {a2n 1} diverges, then {an} diverges. Use these results in this exercise. Consider the alternating series k=1(1)k+1ak, where {4k+1ifkisodd2kifkiseven. a. Show that {an} converges to 0. b. Show that S2n = k=1n1k and explain why limkS2n = . c. Explain why the series k=1(1)k+1ak diverges even though {an} converges to 0. Explain why this result does not contradict the Alternating Series Test.A diverging alternating series Consider the alternating series k=1(1)k+1ak=11211+12212+13213+. a. Show that the individual terms of the series converge to 0. (Hint: See Exercise 70.) b. Find a formula for S2n. the sum of the first 2n terms of the series. c. Explain why the alternating series diverges even though individual terms of the series converge to 0. Explain why this result does not contradict the Alternating Series Test.Evaluate 10!/9!, (k + 2)!/k!, and k!/(k+ 1)!2QCExplain how the Ratio Test works.Explain how the Root Test works.Evaluate 1000!/998! without a calculator.4ESimplify k!(k+2)!, for any integer k 0.6EWhat test is advisable if a series involves a factorial term?Can the value of a series be determined using the Root Test or the Ratio Test?The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 9. k=1(1)kk!The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 10. k=1(2)kk!The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 11. k=1(1)k+1(10k3+k9k3+k+1)kThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 12. k=1(2kk+1)kThe Ratio Test Use the Ratio Test to determine whether the following series converge. 11. k=1k24kThe Ratio Test Use the Ratio Test to determine whether the following series converge. 12. k=1kk2kThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 15. k=1(1)k+1kekThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 16. k=1(1)kk!kkThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 17. k=1(7)kk2The Root Test Use the Root Test to determine whether the following series converge. 22. k=1(1+3k)k2The Ratio Test Use the Ratio Test to determine whether the following series converge. 15. k=12kk99The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 20. k=1(1)kk!k6The Ratio Test Use the Ratio Test to determine whether the following series converge. 17. k=1(k!)2(2k)!The Ratio Test Use the Ratio Test to determine whether the following series converge. 18. 2+416+881+16256+The Root Test Use the Root Test to determine whether the following series converge. 23. k=1(kk+1)2k2The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 24. k=1(1)k+13k2kkThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 25. k=1(1)k+1(3k2+4k2k2+1)kThe Root Test Use the Root Test to determine whether the following series converge. 24. k=1(1ln(k+1))kThe Root Test Use the Root Test to determine whether the following series converge. 25. 1+(12)2+(13)3+(14)4+The Root Test Use the Root Test to determine whether the following series converge. 26. k=1kekThe Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 29. k=1(1)k+1k2kk!k!The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. 30. k=1(k15k)kExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. n!n! = (2n)!, for all positive integers n. 1. b. (2n)!(2n1)! = 2n. b. If limkakk = 14, then 10ak converges absolutely. c. The Ratio Test is a ways inconclusive when applied to ak, where ak is a nonzero rational function of k.Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 32. k=1(4k3+79k21)kChoose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 33. k=1(1)k2k+19k1Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 34. k=1k!(2k+1)!Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 35. k=1(2k+1)!(k!)2Choose your test Use the test of your choice to determine whether the following series converge. 42. k=1(k22k2+1)kChoose your test Use the test of your choice to determine whether the following series converge. 43. k=1k100(k+1)!Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 38. k=1k3sin(1/k3)Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 39. k=1(1)kk3k8+1Choose your test Use the test of your choice to determine whether the following series converge. 58. k=11(1+p)k,p0Choose your test Use the test of your choice to determine whether the following series converge. 45. k=1(kk1)2kChoose your test Use the test of your choice to determine whether the following series converge. 52. k=1(k!)3(3k)!Choose your test Use the test of your choice to determine whether the following series converge. 55. k=12kk!kkChoose your test Use the test of your choice to determine whether the following series converge. 56. k=1(11k)k2Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 45. k=1(1)kk0.99Choose your test Use the test of your choice to determine whether the following series converge. 66. k=2100kkChoose your test Use the test of your choice to determine whether the following series converge. 69. 11!+42!+93!+164!+Choose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 48. k=1(1)ktan1kChoose your test Use the rest of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge. 49. k=1(1)kk3/2+kConvergence parameter Find the values of the parameter p 0 for which the following series converge. 70. k=21(lnk)pConvergence parameter Find the values of the parameter p 0 for which the following series converge. 71. k=2lnkkpConvergence parameter Find the values of the parameter p 0 for which the following series converge. 72. k=21k(lnk)(lnlnk)p53E54EConvergence parameter Find the values of the parameter p 0 for which the following series converge. 75. k=1kpkk+156E57EA glimpse ahead to power series Use the Ratio Test to determine the values of x 0 for which each series converges. 82. k=1xkk!A glimpse ahead to power series Use the Ratio Test to determine the values of x 0 for which each series converges. 83. k=1xk60E61E62E63E64EEvaluate the geometric series in Example 1. Does the alternating series in Example 1 converge?Show the steps in evaluating the limit in Example 2. Example 2 Divergence Test Does the series k=1(110k)k converge or diverge?Verify the limit in the Ratio Test in Example 4. Example 4 More than one Test Does the series k=1k2e2k converge or diverge?Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 1. k=1(1)k(2+1k2)kChoosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 2. k=32k2k2Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 3. k=32k2k2k2Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 4. k=31kln7kChoosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 5. k=101(k9)5Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 6. k=10100kk!k2Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 7. k=1(k2k4+k3+1)Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 8. k=1(3)k4k+1Choosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 9. k=1(1)k+12k+lnkChoosing convergence tests Identify a convergence test for each of the following series. If necessary, explain now to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test. 10. k=1(tan12ktan1(2k2))Applying convergence tests Determine whether the following series converge. Justify your answers. 11. k=12k4+k4k48kApplying convergence tests Determine whether the following series converge. Justify your answers. 12. k=172kApplying convergence tests Determine whether the following series converge. Justify your answers. 13. k=352+lnkApplying convergence tests Determine whether the following series converge. Justify your answers. 14. k=17k2k254k43k+1Applying convergence tests Determine whether the following series converge. Justify your answers. 15. k=1(7)kk!Applying convergence tests Determine whether the following series converge. Justify your answers. 16. k=17kk!+10Applying convergence tests Determine whether the following series converge. Justify your answers. k=1(k)33k3+2Applying convergence tests Determine whether the following series converge. Justify your answers. k=1(1+ak)k, a is realApplying convergence tests Determine whether the following series converge. Justify your answers. k=03k+45k2Applying convergence tests Determine whether the following series converge. Justify your answers. 20. j=14j10j11+1Applying convergence tests Determine whether the following series converge. Justify your answers. 21. k=1(1)kkk3+1Applying convergence tests Determine whether the following series converge. Justify your answers. 22. k=1(e+1)kApplying convergence tests Determine whether the following series converge. Justify your answers. 23. k=1k55kApplying convergence tests Determine whether the following series converge. Justify your answers. 24. k=14(k+3)3Applying convergence tests Determine whether the following series converge. Justify your answers. 25. k=11kekApplying convergence tests Determine whether the following series converge. Justify your answers. 26. k=15+sinkkApplying convergence tests Determine whether the following series converge. Justify your answers. 27. k=13+cos5kk3Applying convergence tests Determine whether the following series converge. Justify your answers. 28. k=3(1)klnkk1/3Applying convergence tests Determine whether the following series converge. Justify your answers. 29. k=110k+1k10Applying convergence tests Determine whether the following series converge. Justify your answers. 30. k=31k3lnkApplying convergence tests Determine whether the following series converge. Justify your answers. 31. j=15j2+4Applying convergence tests Determine whether the following series converge. Justify your answers. 32. k=1kk(k+2)kApplying convergence tests Determine whether the following series converge. Justify your answers. 33. k=31k1/3lnkApplying convergence tests Determine whether the following series converge. Justify your answers. 34. k=1(1)k5k23k5+1Applying convergence tests Determine whether the following series converge. Justify your answers. 35. k=12k3kkkApplying convergence tests Determine whether the following series converge. Justify your answers. 36. k=1(1)k+12k+1k!Applying convergence tests Determine whether the following series converge. Justify your answers. 37. k=1(1)k(5k3k+7)kApplying convergence tests Determine whether the following series converge. Justify your answers. 38. k=12k(k1)!k!Applying convergence tests Determine whether the following series converge. Justify your answers. 39. k=15k(k!)2(2k)!Applying convergence tests Determine whether the following series converge. Justify your answers. 40. j=1cos((2j+1))j2+1Applying convergence tests Determine whether the following series converge. Justify your answers. 41. k=12k3k2kApplying convergence tests Determine whether the following series converge. Justify your answers. 42. k=12+(1)k2kApplying convergence tests Determine whether the following series converge. Justify your answers. 43. k=1cos(3k1)3Applying convergence tests Determine whether the following series converge. Justify your answers. 44. k=101k(k9)Applying convergence tests Determine whether the following series converge. Justify your answers. 45. k=1k4ek5Applying convergence tests Determine whether the following series converge. Justify your answers. 46. k=11(k+1)!k!Applying convergence tests Determine whether the following series converge. Justify your answers. 47. k=1(4k)!(k!)4Applying convergence tests Determine whether the following series converge. Justify your answers. 48. 23+38+415+524+635+..Applying convergence tests Determine whether the following series converge. Justify your answers. 49. k=1k5k7+15Applying convergence tests Determine whether the following series converge. Justify your answers. 50. k=1ek3Applying convergence tests Determine whether the following series converge. Justify your answers. 51. k=17k+11k11kApplying convergence tests Determine whether the following series converge. Justify your answers. 52. k=17k+11k13kApplying convergence tests Determine whether the following series converge. Justify your answers. 53. k=1sin1k9Applying convergence tests Determine whether the following series converge. Justify your answers. 54. j=1j9sin1j9Applying convergence tests Determine whether the following series converge. Justify your answers. 55. k=1cos1k9Applying convergence tests Determine whether the following series converge. Justify your answers. 56. k=1(kk+10)k2Applying convergence tests Determine whether the following series converge. Justify your answers. 57. k=1512kApplying convergence tests Determine whether the following series converge. Justify your answers. 58. k=1(1)kln(k+2)Applying convergence tests Determine whether the following series converge. Justify your answers. 59. k=1k!kk+3Applying convergence tests Determine whether the following series converge. Justify your answers. 60. k=1(1k2+1k5)Applying convergence tests Determine whether the following series converge. Justify your answers. 61. k=11ln(ek+1)Applying convergence tests Determine whether the following series converge. Justify your answers. 62. k=0(tan1k)kApplying convergence tests Determine whether the following series converge. Justify your answers. 63. k=1(k+ak)k2,a0Applying convergence tests Determine whether the following series converge. Justify your answers. 64. 114+127+1310+1413+...Applying convergence tests Determine whether the following series converge. Justify your answers. 65. k=1(cos1kcos1k+1)Applying convergence tests Determine whether the following series converge. Justify your answers. 66. k=14k2k!Applying convergence tests Determine whether the following series converge. Justify your answers. 67. j=1cot11j2jApplying convergence tests Determine whether the following series converge. Justify your answers. 68. k=2k+1kk2kApplying convergence tests Determine whether the following series converge. Justify your answers. 69. k=1(1+12k)kApplying convergence tests Determine whether the following series converge. Justify your answers. 70. k=0ek/100Applying convergence tests Determine whether the following series converge. Justify your answers. 71. k=1ln2kk3/2Applying convergence tests Determine whether the following series converge. Justify your answers. 72. k=1(1)kcos1kApplying convergence tests Determine whether the following series converge. Justify your answers. 73. k=0k21.001kApplying convergence tests Determine whether the following series converge. Justify your answers. 74. k=0k0.999kApplying convergence tests Determine whether the following series converge. Justify your answers. 75. k=1(1k)kApplying convergence tests Determine whether the following series converge. Justify your answers. 76. k=32k(2k)Applying convergence tests Determine whether the following series converge. Justify your answers. 77. k=03kk4+34Applying convergence tests Determine whether the following series converge. Justify your answers. 78. k=0k3k2Applying convergence tests Determine whether the following series converge. Justify your answers. 79. k=1tan11kApplying convergence tests Determine whether the following series converge. Justify your answers. 80. k=1(3k823k9+2)kApplying convergence tests Determine whether the following series converge. Justify your answers. 81. k=1(1k+21k)Applying convergence tests Determine whether the following series converge. Justify your answers. 82. k=0((23)k+1(32)k+1)