Construct the cluster set and audit a sparse power series
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 110 minutes, as an unvalidated planning estimate. This collection contains 2 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Distinguish finite observed maxima from suprema of infinite tails.
Construct strictly increasing subsequence indices and state the domains of extended-real expressions.
Analyze power-series endpoints separately from the radius theorem.
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Form one real sequence by concatenating blocks indexed by k=1,2,3,… . Block k is −1, followed by 1,1/2,…,1/k, followed by 2. Describe and prove the entire set of finite subsequential limits. Compute the lower and upper limits from the infinite tails. Does every number between the lower and upper limits have to be a subsequential limit? Finally let b_n=−a_n for this sequence. Compute limsup b_n and limsup(a_n b_n), and use them to assess the proposed identity limsup(a_n b_n)=(limsup a_n)(limsup b_n).
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For n≥1 define c_n=5^n/n when n is a perfect cube, and c_n=(−2)^n/n² otherwise. Study Σ c_n x^n for real x. Prove that |c_n|^(1/n) has no ordinary limit, find its upper limit, and determine the radius of convergence. Establish what happens at both real endpoints without inferring the answer from the radius alone. Now replace just the cube-index coefficients by 5^n. Which conclusions change?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Form one real sequence by concatenating blocks indexed by k=1,2,3,… . Block k is −1, followed by 1,1/2,…,1/k, followed by 2. Describe and prove the entire set of finite subsequential limits. Compute the lower and upper limits from the infinite tails. Does every number between the lower and upper limits have to be a subsequential limit? Finally let b_n=−a_n for this sequence. Compute limsup b_n and limsup(a_n b_n), and use them to assess the proposed identity limsup(a_n b_n)=(limsup a_n)(limsup b_n).
For n≥1 define c_n=5^n/n when n is a perfect cube, and c_n=(−2)^n/n² otherwise. Study Σ c_n x^n for real x. Prove that |c_n|^(1/n) has no ordinary limit, find its upper limit, and determine the radius of convergence. Establish what happens at both real endpoints without inferring the answer from the radius alone. Now replace just the cube-index coefficients by 5^n. Which conclusions change?
