Faculty Key · Endpoint Testing & Interval of Convergence
Complete solutions, method expectations, validity conditions, and verification guidance.
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Khaemenes Higher Learning
Faculty Key · Endpoint Testing & Interval of Convergence
Show method selection, derivation, conditions, units, verification, and interpretation.
U13 · L02
Name
Date
1
State the central method of Endpoint Testing & Interval of Convergence and list the conditions that must be checked.
Faculty solution: Use testing boundary points separately after finding radius. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Find interval for Σ(x−2)^n/n.
Faculty solution: R=1; at x=3 harmonic diverges, at x=1 alternating converges, so [1,3).
3
Find interval for Σx^n/n².
Faculty solution: R=1 and both endpoints converge absolutely: [−1,1].
4
A learner makes an error by using a power or Taylor series without interval, center, endpoint, or error analysis. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is using a power or Taylor series without interval, center, endpoint, or error analysis. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of Endpoint Testing & Interval of Convergence in scientific approximation, local models, differential equations, computation, and advanced-study preparation. 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 scientific approximation, local models, differential equations, computation, and advanced-study preparation.
6
Describe two independent ways to verify a result from Endpoint Testing & Interval of Convergence.
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.