Alternating Series

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# Alternating Series - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Alternating Series' - robert

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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 …

• Must converge if
• { ak } is a decreasing sequence(that is ak + 1 ≤ ak for all k )
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
• In checking for l'Hopital's rule may be useful
• Consider
• Find
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!
• Show that this alternating series converges absolutely
• Hint: recall rules about p-series
Conditional Convergence
• It is still possible that even thoughdiverges …
• can still converge
• This is called conditional convergence
• Example – consider vs.
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
• Given the following alternating series
• Use generalized ratio test
Assignment
• Lesson 9.5
• Page 636
• Exercises 1 – 29 EOO