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Khaemenes Higher Learning
Faculty Key · The nth-Term Divergence Test
Show method selection, derivation, conditions, units, verification, and interpretation.
U11 · L05
Name
Date
1
State the central method of The nth-Term Divergence Test and list the conditions that must be checked.
Faculty solution: Use using the necessary condition a_n→0. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Determine Σ n/(n+1).
Faculty solution: Diverges because terms approach 1, not 0.
3
Can lim a_n=0 prove convergence?
Faculty solution: No; the harmonic series is a counterexample.
4
A learner makes an error by confusing term limits with series convergence. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is confusing term limits with series convergence. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of The nth-Term Divergence Test in repeated processes, finance, approximation, fractals, and infinite accumulation. 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 repeated processes, finance, approximation, fractals, and infinite accumulation.
6
Describe two independent ways to verify a result from The nth-Term Divergence 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.