Complete solutions, method expectations, validity conditions, and verification guidance.
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Khaemenes Higher Learning
Faculty Key · Limit Comparison Test
Show method selection, derivation, conditions, units, verification, and interpretation.
U12 · L02
Name
Date
1
State the central method of Limit Comparison Test and list the conditions that must be checked.
Faculty solution: Use comparing asymptotic size when direct inequalities are awkward. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Test Σ(3n+1)/(n³+2).
Faculty solution: Compare with 1/n²; limit is 3, so it converges.
3
Test Σn/(n²+1).
Faculty solution: Compare with 1/n; limit is 1, so it diverges.
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 Limit Comparison Test 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 Limit Comparison Test.
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.