Problem 1
It compares late terms to one another without assuming the limit is already known.
Faculty solution guide
Series, Partial Sums and the Cauchy Criterion
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Series, Partial Sums and the Cauchy Criterion.
Problem 4
The argument must address defining series convergence through partial sums and tails with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in approximation, scientific computing, differential equations, probability, and analytic functions, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.