--- name: math-thumb-analysis description: "Apply Chapter 5 (Analysis: Know When Limits and Proofs Are Safe) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to write the hypotheses that make a convergence or stopping claim defensible." --- # Analysis: Know When Limits and Proofs Are Safe Use this chapter to help the reader make a checked mathematical decision. All 12 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Domain and metric; quantifiers; convergence notion; uniform bounds; completeness or compactness assumptions. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Is the problem about perturbation or numerical error?** Find a Lipschitz or derivative bound. If an iteration is a verified contraction, translate its last step into a stopping certificate. - **Are functions converging?** Before moving a limit through continuity, integration, differentiation, or summation, name the governing interchange theorem. Check whether its control is uniform and whether one bound works across the entire domain. - **Is the object a function series?** Seek a point-independent majorant and try the M-test. - **Is the limit unknown?** Compare late terms or partial sums with one another. A tail estimate plus completeness may be enough. - **Does a numerical series change sign?** Test absolute convergence first. If that fails, move to cancellation-sensitive tests. - **Does a positive term have a recognizable leading scale?** Compare it with a $p$-series, geometric series, or logarithmic benchmark. - **Is a proof stalled?** Inspect its architecture: use density plus continuity for identity, formal negation for counterexamples, a stronger invariant for induction, compactness for existence, or backward error algebra for epsilon-delta construction. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Finite numerical checks cannot prove a universal quantified assertion. Name the theorem and verify its domain-wide hypotheses. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 5.1.2, 5.1.3, 5.3.2 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C05-D01: Turn convergence speed into a stopping certificate**: rule 5.1.2. - **C05-D02: Watch a boundary layer defeat uniform convergence**: rule 5.1.3. - **C05-D03: Control a whole function series at once**: rule 5.2.1. - **C05-D04: Test absolute convergence before reordering a series**: rule 5.2.3. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.