Series ConvergenceDivergence Flow Chart TEST FOR DIVERGENCE Does lim   Diverges NO SERIES Does   n  YES Is p   YES Converges YES Diverges NO GEOMETRIC SERIES Does ar  NO Is  YES  YES Diverges NO ALTE
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Series ConvergenceDivergence Flow Chart TEST FOR DIVERGENCE Does lim Diverges NO SERIES Does n YES Is p YES Converges YES Diverges NO GEOMETRIC SERIES Does ar NO Is YES YES Diverges NO ALTE

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Series ConvergenceDivergence Flow Chart TEST FOR DIVERGENCE Does lim Diverges NO SERIES Does n YES Is p YES Converges YES Diverges NO GEOMETRIC SERIES Does ar NO Is YES YES Diverges NO ALTE




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Presentation on theme: "Series ConvergenceDivergence Flow Chart TEST FOR DIVERGENCE Does lim Diverges NO SERIES Does n YES Is p YES Converges YES Diverges NO GEOMETRIC SERIES Does ar NO Is YES YES Diverges NO ALTE"— Presentation transcript:


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Series Convergence/Divergence Flow Chart TEST FOR DIVERGENCE Does lim = 0? Diverges NO -SERIES Does = 1 /n 1? YES Is p > 1? YES Converges YES Diverges NO GEOMETRIC SERIES Does ar 1? NO Is 1? YES =1 YES Diverges NO ALTERNATING SERIES Does = ( 1) or = ( 1) 0? NO Is +1 & lim = 0? YES Converges YES TELESCOPING SERIES Do subsequent terms cancel out previous terms in the sum? May have to use partial fractions, properties of logarithms, etc. to put into appropriate form. NO Does lim finite? YES YES Diverges NO TAYLOR SERIES Does NO Is in interval of convergence? YES =0 YES Diverges NO Try one or more of the following tests: NO COMPARISON TEST Pick . Does converge? Is 0 YES Converges YES Is 0 NO NO Diverges YES LIMIT COMPARISON TEST Pick . Does lim c > finite & , b 0? Does =1 converge? YES Converges YES Diverges NO INTEGRAL TEST Does ), ) is contin- uous, positive & decreasing on a, )? Does dx converge? YES Converges YES Diverges NO RATIO TEST Is lim +1 /a |6 = 1? Is lim +1 1? YES Abs. Conv. YES Diverges NO ROOT TEST Is lim = 1? Is lim 1? YES Abs. Conv. YES Diverges NO
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Problems 1-38 from Stewart’s Calculus , page 784 1. =1 2. =1 3. =1 4. =1 1) 5. =1 3) +1 6. =1 1 + 8 7. =2 ln( 8. =1 + 2)! 9. =1 10. =1 11. =2 1) +1 ln( 12. =1 1) + 25 13. =1 14. =1 sin( 15. =0 (3 + 2) 16. =1 + 1 + 1 17. =1 1) /n 18. =2 1) 19. =1 1) ln( 20. =1 + 5 21. =1 2) 22. =1 + 2 + 5 23. =1 tan(1 /n 24. =1 cos( n/ 2) + 4 25. =1 26. =1 + 1 27. =1 ln( + 1) 28. =1 /n 29. =1 tan 30. =1 1) + 5 31. =1 + 4 32. =1 (2 33. =1 sin(1 /n 34. =1 cos 35. =1 + 1 36. =2 (ln( )) ln( 37. =1 1) 38. =1 1)