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Calculus II practice

Sequence Limits and Partial Sums worksheets

Practice explicit and recursive limits, finite partial sums, divergence, and monotone-bounded reasoning.

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Practice explicit and recursive limits, finite partial sums, divergence, and monotone-bounded reasoning.

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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.

Learning objective

What this activity builds

Distinguish a sequence limit from a finite or infinite series conclusion.

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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.

Teaching notes

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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 SheetNameScoreTeacherDateSequence Limits and Partial SumsDistinguish sequence limits from series partial sums. Show divergence explicitly and justify monotone/bounded conclusions.1.Let a_1=-5 and a_(n+1)=(a_n+0)/2. Prove the sequence is monotone and bounded, then state its limit.2.Let a_1=21 and a_(n+1)=(a_n+7)/2. Determine lim_(n->infinity) a_n.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateSequence Limits and Partial SumsDistinguish sequence limits from series partial sums. Show divergence explicitly and justify monotone/bounded conclusions.3.For a_n = 1 + 4(-1)^n, determine lim_(n->infinity) a_n. State whether the sequence converges or diverges.4.For the recursive sequence a_1=6, a_(n+1)=(-1/4)a_n, find the finite partial sum S_8=a_1+...+a_8. Do not replace it with the infinite-series sum.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateSequence Limits and Partial SumsDistinguish sequence limits from series partial sums. Show divergence explicitly and justify monotone/bounded conclusions.5.For the explicit sequence a_n=8(1/2)^(n-1), find the finite partial sum S_7=a_1+...+a_7. Do not replace it with the infinite-series sum.6.For a_n=3 + 8/n^3, decide whether the sequence is increasing/decreasing and bounded. Then state its limit.Separate the object, test, endpoint, approximation, and conclusion.math-sheet.comPage 3 of 3

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