1 / 13

Converges by Integral Test

p-series with p=2>1 converges. Converges by DCT. Larger Denom makes fraction smaller. Diverges by the n th term test (Lim ≠ 0 then it diverges). Continuous if n>1 Positive if n>1 Decreasing if n>1. p-series with p = 7/2 > 1 Converges. Converges by Integral Test.

Download Presentation

Converges by Integral Test

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. p-series with p=2>1 converges Converges by DCT Larger Denom makes fraction smaller Diverges by the nth term test (Lim ≠ 0 then it diverges) Continuous if n>1 Positive if n>1 Decreasing if n>1 p-series with p = 7/2 > 1 Converges Converges by Integral Test

  2. Chapter 9(5)Alternating Series TestAlternating Series RemainderAbsolute and Conditional ConvergenceRearranging an infinite series

  3. Alternating series contain both positive and negative terms – the signs alternate This is an Alternating Geometric series Alternating Series Test If terms are positive, limit = 0, and terms get smaller, then an alternating series converges

  4. Determine convergence or divergence of: Check an > 0 Check Lim = 0 Check terms get smaller The series converges by the Alternating Series Test (AST)

  5. Can the AST be used to show convergence or divergence of: Check an > 0 Check Lim = 0 Check terms get smaller This part fails The Alternating Series Test (AST) can not be applied

  6. Alternating Series Remainder For a convergent alternating series, a range for the sum can be found by taking n terms, finding the remainder and using it to establish a range

  7. Approximate the sum from the first six terms: Check an > 0 Check Lim = 0 Check terms get smaller Series converges by the (AST)

  8. Absolute Convergence

  9. Describe the convergence of each: Check: an > 0 Lim = 0 an+1<an Not Decreasing so the AST Fails Check Absolute Value Converges by AST Divergent p-series (p < 1) Conditional Convergence

  10. Rearrangement of a series If you rearrange a finite series the sum does not change Absolute converging series can also be rearranged with no change

  11. Conditional converging series can be rearranged to change the sum Consider: an Positive? Lim = 0? an+1 < an ? Converges by the AST This is the harmonic series which diverges

  12. We have a conditional converging series (Proved later) Evens are negative Put fractions together with double denominators Combine insides Factor out 1/2 From initial series

  13. Geometric series with r = ½ < 1 Diverges by the nth term test (Lim ≠ 0 then it diverges) Converges Conditional or absolute? p-series with p = ½ < 1 Diverges The series converges p-series with p = ½ < 1 Diverges The series converges conditionally

More Related