Study Guide - Test 3 (MATH-2212, Spring-2017).pdf -...

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MATH-2212: Calculus of One Variable IIStudy Guide for Test 3, Spring 2017Study Guide for Test 3It goes without saying that mathematics develops cumulatively, and all subsequent topics usuallydepend heavily on some (if not all) previous material. This study guide is intended to emphasizethe new material learned in this portion of the course. Knowing and understanding everything thathas been covered before — in prerequisite courses and earlier in this course — is also crucial forsuccess on this test.Test ContentsSection 11.2– SeriesSection 11.3– The Integral Test and Estimates of SumsSection 11.4– The Comparison TestsSection 11.5– Alternating SeriesSection 11.6– Absolute Convergence and the Ratio and Root TestsSection 11.7– Strategy for Testing SeriesKey ConceptsInfinite series, partial sums, the sum of an infinite series, convergence and divergence of infiniteseries, remainders of an infinite series.Some special types of series: telescoping series, geometric series, harmonic series,p-series.Arithmetic properties of convergent series.The Test for Divergence.The Integral Test and its remainder estimate.The Direct Comparison Test and the Limit Comparison Test.The Alternating Series Test and its remainder estimate.The Ratio Test and the Root Test.Absolute and conditional convergence of series with mixed (positive and negative) terms;examining series for absolute or conditional convergence or divergence.– 1–
MATH-2212: Calculus of One Variable IIStudy Guide for Test 3, Spring 2017Tests for SeriesTest for Divergence.Given a seriesan.If limn→∞an6= 0 (or DNE), then the series diverges.If limn→∞an= 0, then inconclusive.Geometric Series.It is a series of the formn=0arn=a+ar+ar2+ar3+· · ·.If|r|<1, then the series converges, andn=0arn=a1-r. (Hereais the initial term ofthe geometric series.)If|r| ≥1, then the series diverges.p-Series.It is a series of the form1np.Ifp >1, then the series converges.Ifp1, then the series diverges.Integral Test.Letf(x) be a function, and consider the seriesn=Nf(n).

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