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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.

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Converges by Integral Test

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  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

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