Complete solutions, method expectations, validity conditions, and verification guidance.
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Khaemenes Higher Learning
Faculty Key · Direct Comparison Test
Show method selection, derivation, conditions, units, verification, and interpretation.
U12 · L01
Name
Date
1
State the central method of Direct Comparison Test and list the conditions that must be checked.
Faculty solution: Use building inequalities against known benchmark series. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Determine Σ1/(n²+3).
Faculty solution: Converges by comparison with 1/n².
3
Determine Σ1/(2n+1).
Faculty solution: Diverges by comparison with a constant multiple of 1/n.
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 Direct 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 Direct 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.