Faculty Key · Test Selection, Counterexamples & Decision Logic
Complete solutions, method expectations, validity conditions, and verification guidance.
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Khaemenes Higher Learning
Faculty Key · Test Selection, Counterexamples & Decision Logic
Show method selection, derivation, conditions, units, verification, and interpretation.
U12 · L07
Name
Date
1
State the central method of Test Selection, Counterexamples & Decision Logic and list the conditions that must be checked.
Faculty solution: Use selecting a test from structure and explaining inconclusive outcomes. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Why is the ratio test inconclusive for Σ1/n²?
Faculty solution: The ratio limit is 1.
3
Name a useful first question when testing a series.
Faculty solution: Do the terms approach zero? If not, the series diverges immediately.
4
A learner makes an error by using a convergence test outside its hypotheses or treating an inconclusive result as a conclusion. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is using a convergence test outside its hypotheses or treating an inconclusive result as a conclusion. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of Test Selection, Counterexamples & Decision Logic in error control, stability, perturbation methods, and approximation reliability. Define variables and identify the quantity an integral, differential equation, sequence, or series would measure.
Faculty solution: A defensible response defines variables and units, states assumptions, builds the correct mathematical object, and interprets its value in error control, stability, perturbation methods, and approximation reliability.
6
Describe two independent ways to verify a result from Test Selection, Counterexamples & Decision Logic.
Faculty solution: Use an exact structural check such as differentiation, substitution, partial sums, or theorem hypotheses; then compare with numerical, graphical, dimensional, or limiting evidence.