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f j: fl4 c&b& I\Iatb 1320.001 Quiz 06 Fall 2015 Class ID: Name Write your 3 favorite convergence tests for series in the space provided below. Your descrip tioii should be of the form “Given.., if... then...” Exairiple: Root Test Given a series of the form a, /[[ < 1, then the series converges absolutely. if iim /JJ > 1, then the series diverges. if 1im 11 /‘fJ 1, then the test is inconclusive. (i) if lim÷ (ii) (iii) = You may dusuribe any of time tests frommi the flow chart except the root test given above. 1. Test: -ftix cLt 2. Test: 3. Test: / // SERIES CONVERGENCE/DIVERGENCE FLOW CHART TEST FOR DIVERGENCE Does li,_ a,, = 0? YES p-SERIES Does a = 1/oF, n NO verges 1? YES NO H GEOMETRIC SERIES Does a = ar” , > 1? 1 YES NO I 1. ALTERNATING SERIES Does a,, = (—1)b,, or 0? = (—1)”’b,,, b,, b & linb 1 YESHIS b,,÷ = 0? 5YES -j TELES COPING SERIES Do subsequent terms cancel out previous terms in he sum? May have to use partial fractions, properties of logat itlun etc to put into appi opi iate foi in i— YES — Does liniS YES S - s finite” NO TAYLOR SERIES Does a, = fC”)(a) (i — YES a)”? —‘i Is x in interval of convergence? YES NO 1, Try one or more of the following tests IsO <a <ba? COMPARISON TEST nverges YES IsO<b<a? LIMIT COMPARISON TEST H Pick{b,,}. Does lim c finite & a,,,b,, >0? INTEGRAL TEST Does a,, = f(n), f(x) is contin uous, positive & decreasing on [a,oo)? RATIO TEST Is lim a+i/aI H ROOT TEST Is 1im /i H 1? 1? — YES H YES y anonverges Does f f(x)di converge? ft YESNO