Show method selection, derivation, conditions, units, verification, and interpretation.
U11 · L02
Name
Date
1
State the central method of Monotone, Bounded & Recursive Sequences and list the conditions that must be checked.
Faculty solution: Use using structural properties to prove convergence. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Is a_n=1−1/n increasing and bounded?
Faculty solution: Yes; it increases and is bounded above by 1, so it converges to 1.
3
For a_{n+1}=(a_n+2/a_n)/2 with a₁>0, what quantity is approximated?
Faculty solution: √2.
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 Monotone, Bounded & Recursive Sequences 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 Monotone, Bounded & Recursive Sequences.
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.