State the central method of Ratio Test and list the conditions that must be checked.
Faculty solution: Use measuring factorial and exponential growth. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
Unit 12 · Week 34 · Faculty Key 05
Faculty Key · Ratio Test
Complete solutions, method expectations, validity conditions, and verification guidance.
Khaemenes Higher Learning
Faculty Key · Ratio Test
Show method selection, derivation, conditions, units, verification, and interpretation.
Name
Date
Test Σ3^n/n!.
Faculty solution: L=0, so it converges absolutely.
Test Σn!/2^n.
Faculty solution: The ratio tends to infinity, so it diverges.
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.
Create a short application of Ratio 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.
Describe two independent ways to verify a result from Ratio 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.