Alternating Series

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

Alternating Series