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MTH162 - Chapter 8

Bob Harold Englisch 95 Karten

MTH162 - Chapter 8

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  1. 18.1 Sequences: What is a sequence?a list of numbers written in a specific order
    Represented by "asubn", it could be a function.
    Those functions generate terms
  2. 28.1 Sequences: What is a sequence? asubn from n=0 to ∞{a₀, a₁, a₂,…, asubn, asubn+1,…}
  3. 38.1 Sequences: How to graph the sequence of asubnplot the points (n, asubn)) as n ranges over all possible values on a graph8.1 Sequences: How to graph the sequence of asubn
  4. 48.1 Sequences: that begin with positive termscontains a function with
    (-1)ⁿ⁺¹
  5. 58.1 Sequences: that begin with negative termscontains a function with
    (-1)ⁿ
  6. 68.1 Sequences: for "∑" if "n" is not specified...use n=1
  7. 78.1 Sequences: asubnIs typically a function.8.1 Sequences: asubn
  8. 88.1 Sequences: lim n→∞ asubn exists and is finite, the sequence is..Convergent
  9. 98.1 Sequences: lim n→∞ asubn doesnt exist and is infinite, the sequence is..Divergent
  10. 108.1 Sequences: lim n→∞ asubn Divergent sequences =−∞
    ∞
  11. 118.1 Sequences:8.1 Sequences:8.1 Sequences:
  12. 128.1 Sequences:8.1 Sequences:8.1 Sequences:
  13. 138.1 Sequences:8.1 Sequences:8.1 Sequences:
  14. 148.1 Sequences:8.1 Sequences:8.1 Sequences:
  15. 158.1 Sequences:8.1 Sequences:8.1 Sequences:
  16. 168.1 Sequences: the is sequence increasing ifasubn < asub(n+1) for every n.
    (each term is larger than the one before)
  17. 178.1 Sequences: the is sequence decreasing ifasubn > asub(n+1) for every n.
    (each term is smaller than the one before)
  18. 188.1 Sequences: n!n*(n-1)*(n-2)*(n-3)...until n=1
    Ex: 4! = 4*3*2*1
  19. 198.1 Sequences: Series Convergence Definition8.1 Sequences: Series Convergence Definition
  20. 208.2 Series What is a series?Sum of the terms of a sequence.8.2 Series What is a series?
  21. 218.2 Series Goals of all series teststo find convergence or divergence
  22. 228.2 Series nth Partial Sum8.2 Series nth Partial Sum
  23. 238.2 Series Telescoping Series 1.18.2 Series Telescoping Series 1.1
  24. 248.2 Series Telescoping Series 1.2Make sure you use
    Partial Fractions on the function:
    2 = a/(n+1) + b/(n-1)
    use values of a and b to construct formula for the series nth Partial Sum
  25. 258.2 Series Telescoping Series 1.3 Partial Fractions Example to construct formula for the series nth Partial Sum8.2 Series Telescoping Series 1.3 Partial Fractions Example to construct formula for the series nth Partial Sum
  26. 268.2 Series Telescoping Series 2 Simpler Example8.2 Series Telescoping Series 2 Simpler Example
  27. 27List all tests and their convergence and divergences conditionsTest for Divergence
    Geometric series Test
    Integral Test
    P series
    Direct comparison Test
    Limit comparison test
    Alternating series test
    Ratio Test
  28. 288.2 Series Test for Divergence 18.2 Series Test for Divergence 18.2 Series Test for Divergence 1
  29. 298.2 Series Test for Divergence 28.2 Series Test for Divergence 2
  30. 308.2 Series Test for Divergence Example 18.2 Series Test for Divergence Example 1
  31. 318.2 Series Test for Divergence Example 28.2 Series Test for Divergence Example 2
  32. 328.2 Series Test for Divergence Example 38.2 Series Test for Divergence Example 3
  33. 338.2 Series Test for Divergence Example 48.2 Series Test for Divergence Example 4
  34. 348.2 Series Test for Divergence Example 58.2 Series Test for Divergence Example 5
  35. 358.2 Series Test for Divergence Example 68.2 Series Test for Divergence Example 6
  36. 368.2 Series Test for Divergence Example 78.2 Series Test for Divergence Example 7
  37. 378.2 Series Geometric Series Divergesif r is not -1 < |r| < 1
  38. 388.2 Series Geometric Series Converges-1 < |r| < 1
  39. 398.2 Series Geometric Series formula (two forms)8.2 Series Geometric Series formula (two forms)
  40. 408.2 Series Geometric Series Example 18.2 Series Geometric Series Example 1
  41. 418.2 Series Geometric Series Example 28.2 Series Geometric Series Example 2
  42. 428.3 Tests Integral Test ConvergentSeries is Convergent if integral is Convergent.8.3 Tests Integral Test Convergent
  43. 438.3 Tests Integral Test DivergentSeries is Divergent if integral is Divergent.8.3 Tests Integral Test Divergent
  44. 448.3 Tests Integral Test Is it the answer to the series?No.
  45. 458.3 Tests P-series Formula8.3 Tests P-series Formula
  46. 468.3 Tests P-series Formula reconsidered8.3 Tests P-series Formula reconsidered
  47. 478.3 Tests P-series Convergentp > 1
  48. 488.3 Tests P-series Divergentp ≤ 1
  49. 498.3 Tests P-series is result of set of rules derived from....integral tests. If you do an Integral test on p-series formula, you will get the same resul.
  50. 508.3 Tests NOT a P-series8.3 Tests NOT a P-series
  51. 518.3 Tests P-series Breaking down a series into a p-series8.3 Tests P-series Breaking down a series into a p-series
  52. 528.3 Tests Direct Comparison Test - What kind of sequences apply? - asubn ≤ or ≥ bsubn for all n?8.3 Tests Direct Comparison Test - What kind of sequences apply? - asubn ≤ or ≥ bsubn for all n?
  53. 538.3 Tests Direct Comparison Test if bsubn convergesso does asubn
  54. 548.3 Tests Direct Comparison Test if asubn divergesso does bsubn
  55. 558.3 Tests Direct Comparison Test Example8.3 Tests Direct Comparison Test Example
  56. 568.3 Tests Direct Comparison Test if bsubn divergesasubn still might be convergent
    Use limit comparison test
  57. 578.3 Tests Direct Comparison Test if bsubn diverges ExampleDirect Comparison test fails.
    Use limit comparison test.    8.3 Tests Direct Comparison Test if bsubn diverges Example
  58. 588.3 Tests Limit Comparison Test Definition8.3 Tests Limit Comparison Test Definition
  59. 598.3 Tests Limit Comparison test Example 18.3 Tests Limit Comparison test Example 1
  60. 608.3 Tests Limit Comparison test Example 28.3 Tests Limit Comparison test Example 2
  61. 618.4 More Convergence Tests Alternating Series Test Contains (-1)ⁿ⁻¹Example: (-1)ⁿ⁻¹ * asubn
    Begins series as positive
    (+)+(-)+(+)+(-)+(+)+(-)....
  62. 628.4 More Convergence Tests Alternating Series Test Contains (-1)ⁿExample: (-1)ⁿ * asubn
    Begins series as negative
    (-)+(+)+(-)+(+)+(-)....
  63. 638.4 More Convergence Tests Alternating Series Test Convergence Criteria∑ must be positive8.4 More Convergence Tests Alternating Series Test Convergence Criteria
  64. 648.4 More Convergence Tests Alternating Series Test Divergence Criteria≠ convergence criteria
  65. 658.4 More Convergence Tests Alternating Series Test Can Integral Test be used?No, Integral tests can not be used on formulas with this format.8.4 More Convergence Tests Alternating Series Test Can Integral Test be used?
  66. 668.4 More Convergence Tests Alternating Series Test 2 Steps to complete1) Separate the formula into AST form
    (∑(-1)ⁿ⁻¹ asubn or ∑(-1)ⁿ asubn )
    2) FInd lim n→∞ asubn
  67. 678.4 More Convergence Tests Absolutely Convergent∑ must be positive8.4 More Convergence Tests Absolutely Convergent
  68. 688.4 More Convergence Tests Conditionally Convergent∑ must be positive8.4 More Convergence Tests Conditionally Convergent
  69. 698.4 More Convergence Tests Ratio Test Formula8.4 More Convergence Tests Ratio Test Formula
  70. 708.4 More Convergence Tests Ratio Test Convergent8.4 More Convergence Tests Ratio Test Convergent
  71. 718.4 More Convergence Tests Ratio Test Divergent8.4 More Convergence Tests Ratio Test Divergent
  72. 728.4 More Convergence Tests Ratio Test Inconclusive8.4 More Convergence Tests Ratio Test Inconclusive
  73. 738.4 More Convergence Tests Ratio Test Example 18.4 More Convergence Tests Ratio Test Example 1
  74. 748.4 More Convergence Tests Ratio Test Example 28.4 More Convergence Tests Ratio Test Example 2
  75. 758.4 More Convergence Tests Ratio Test (1+1/n)^n =e
  76. 768.5 Power Series formula without centering8.5 Power Series formula without centering
  77. 778.5 Power Series formula WITH centering8.5 Power Series formula WITH centering
  78. 788.5 Power Series 1) radius of convergence = r8.5 Power Series 1) radius of convergence = r
  79. 798.5 Power Series 2) radius of convergence = r-Use the ratio test
    -Isolate x
    -complete the limit
    x/# < 1 then multiply # by 1
  80. 808.5 Power Series 1) Interval of convergence8.5 Power Series 1) Interval of convergence
  81. 818.5 Power Series 1) FULL EXAMPLE8.5 Power Series 1) FULL EXAMPLE
  82. 828.5 Power Series FULL EXAMPLE 18.5 Power Series FULL EXAMPLE 1
  83. 838.5 Power Series FULL EXAMPLE 28.5 Power Series FULL EXAMPLE 2
  84. 848.5 Power Seriesr=18.5 Power Series8.5 Power Series
  85. 858.5 Power Series eˣR=∞8.5 Power Series eˣ
  86. 868.5 Power Series sinxR=∞8.5 Power Series sinx
  87. 878.5 Power Series cosxR=∞8.5 Power Series cosx
  88. 888.5 Power Series ln(1+x)R=18.5 Power Series ln(1+x)
  89. 89limits lim n→∞(aˣ)0
  90. 90limits lim n→∞ (ln(7n+9) - ln(6n-6))∞-∞ = Divergent undefined
  91. 91limitslimitslimits
  92. 92limits lim n→∞ n!/8^n4^nlimits lim n→∞ n!/8^n4^n
  93. 93—
  94. 94—
  95. 95—

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Karteikarten am MTH162 - Chapter 8, erstellt von Bob Harold am 02/04/2021.
Erstellt von
Bob Harold
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