# Summary of Series Tests - Summary of Convergence Tests for...

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Chapter 11 / Exercise 4
Nature of Mathematics
Smith
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Summary of Convergence Tests for Infinite SeriesTestSeriesConverges ifDiverges ifCommentDivergenceXn=1anlimn→∞an6= 0cannot be used to show convergenceGeometric SeriesXn=0arn|r|<1|r| ≥1Sum:S=a1-rp-SeriesXn=11npp >1p1IntegralXn=1anZ1f(x)dxZ1f(x)dxfis continuous, positive,an=f(n)0convergesdivergesand decreasing0anbn0bnanComparisonXn=1anandXn=1bnandXn=1bnan, bnpositiveconvergesdivergesan, bnpositivelimn→∞anbn>0limn→∞anbn>0Test fails ifLimit ComparisonXn=1anandandlimn→∞anbn= 0Xn=1bnconvergesXn=1bndivergesorlimn→∞anbn=Alternating SeriesXn=1(-1)n-1bnbn+1bnlimn→∞bn6= 0bnpositivelimn→∞bn= 0Absolute ConvergenceXn=1anXn=1|an|cannot be used to show divergenceconvergesRatioXn=1anlimn→∞an+1an<1limn→∞an+1an>1Test fails iflim
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Chapter 11 / Exercise 4
Nature of Mathematics
Smith
Expert Verified