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Series Convergence Tests and Error Bounds worksheets

Apply integral, comparison, alternating, root, and ratio tests with absolute/conditional classification and remainder bounds.

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Apply integral, comparison, alternating, root, and ratio tests with absolute/conditional classification and remainder bounds.

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Conclusions are tied to visible reasoning

Generated keys distinguish sequence limits, partial sums, infinite-series convergence, endpoint behavior, and approximation error instead of collapsing them into a single answer.

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Select a valid convergence test and state exactly what it proves.

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The governing limit or test, its hypotheses, the calculation, and a precise conclusion.

The worked key preserves the reasoning that distinguishes similar-looking sequence and series tasks.

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  • Have students name the mathematical object before choosing a method.
  • Require separate endpoint checks for power series.
  • Keep exact values and labeled approximations side by side.

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Math SheetNameScoreTeacherDateSeries Convergence Tests and Error BoundsChoose and apply a valid convergence test, classify absolute, conditional, or divergent behavior, and show the selected remainder bound when requested.1.Use the ratio test as far as it applies to classify sum_(n=1)^infinity n!/8^n as absolute, conditional, or divergent. If it converges, bound |R_13|.2.Use the direct comparison test as far as it applies to classify sum_(n=1)^infinity (-1)^(n+1)/(n+5) as absolute, conditional, or divergent. If itconverges, bound |R_14|.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateSeries Convergence Tests and Error BoundsChoose and apply a valid convergence test, classify absolute, conditional, or divergent behavior, and show the selected remainder bound when requested.3.Use the limit comparison test as far as it applies to classify sum_(n=1)^infinity 1/(n+2) as absolute, conditional, or divergent. If it converges,bound |R_11|.4.Use the alternating test as far as it applies to classify sum_(n=1)^infinity (-1)^(n+1)/(n+7) as absolute, conditional, or divergent. If it converges,bound |R_14|.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateSeries Convergence Tests and Error BoundsChoose and apply a valid convergence test, classify absolute, conditional, or divergent behavior, and show the selected remainder bound when requested.5.Use the root test as far as it applies to classify sum_(n=1)^infinity (1+1/7)^n as absolute, conditional, or divergent. If it converges, bound |R_9|.6.Use the direct comparison test as far as it applies to classify sum_(n=1)^infinity 1/(n+7) as absolute, conditional, or divergent. If it converges,bound |R_14|.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 3 of 3

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