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Alternating Series. Lesson 9.5. Alternating Series. Two versions When odd-indexed terms are negative When even-indexed terms are negative. Alternating Series Test. Recall does not guarantee convergence of the series In case of alternating series …

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alternating series1
Alternating Series

Two versions

  • When odd-indexed terms are negative
  • When even-indexed terms are negative
alternating series test
Alternating Series Test
  • Recall does not guarantee convergence of the series

In case of alternating series …

  • Must converge if
    • { ak } is a decreasing sequence(that is ak + 1 ≤ ak for all k )
alternating series test1
Alternating Series Test
  • Text suggests starting out by calculating
  • If limit ≠ 0, you know it diverges
  • If the limit = 0
    • Proceed to verify { ak } is a decreasing sequence
  • Try it …
using l hopital s rule
Using l'Hopital's Rule
  • In checking for l'Hopital's rule may be useful
  • Consider
  • Find
absolute convergence
Absolute Convergence
  • Consider a series where the general terms vary in sign
    • The alternation of the signs may or may not be any regular pattern
  • If converges … so does
  • This is called absolute convergence
absolutely
Absolutely!
  • Show that this alternating series converges absolutely
  • Hint: recall rules about p-series
conditional convergence
Conditional Convergence
  • It is still possible that even thoughdiverges …
    • can still converge
  • This is called conditional convergence
  • Example – consider vs.
generalized ratio test
Generalized Ratio Test
  • Given
    • ak≠ 0 for k ≥ 0 and
    • where L is real or
  • Then we know
    • If L < 1, then converges absolutely
    • If L > 1 or L infinite, the series diverges
    • If L = 1, the test is inconclusive
apply general ratio
Apply General Ratio
  • Given the following alternating series
    • Use generalized ratio test
assignment
Assignment
  • Lesson 9.5
  • Page 636
  • Exercises 1 – 29 EOO