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Khaemenes Higher Learning
Faculty Key · Alternating Series & Error Bounds
Show method selection, derivation, conditions, units, verification, and interpretation.
U12 · L03
Name
Date
1
State the central method of Alternating Series & Error Bounds and list the conditions that must be checked.
Faculty solution: Use using decreasing magnitudes and a zero limit. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Test Σ(−1)^{n+1}/n.
Faculty solution: Converges by the alternating-series test.
3
How many terms ensure error below .01?
Faculty solution: Need 1/(N+1)<.01, so N≥100.
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 Alternating Series & Error Bounds 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 Alternating Series & Error Bounds.
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.