Let and be sequences of positive terms with for all (for some fixed )
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Let {an} and {bn} be sequences of positive terms with an≤bn for all n≥N (for some fixed N). Consider the two series ∑n=1∞an and ∑n=1∞bn.
(a) Describe, in words, the type of conclusion that can be drawn about ∑an when ∑bn is known to converge and the above inequality holds. Provide a brief rationale without performing any calculations.
(b) Suppose instead that bn≤an for all n≥N and ∑bn is known to diverge. State and explain the corresponding conclusion that can be reached about ∑an.
(c) Explain why the direction of the inequality between an and bn is crucial when using this comparison method. Illustrate your explanation with a hypothetical example (you may invent simple numerical sequences) showing that reversing the inequality can invalidate the conclusion.