Calculus II | Section 8.1 - Q#15: Sequences | Convergence and Divergence

In this lesson, we determine whether a sequence converges or diverges. We examine its numerical pattern, visualize its behavior using graphs, and apply the subsequence theorem to justify our conclusion. PROBLEM: Determine whether the following sequence converges or diverges. If it converges, find its limit. aₙ = {0, 5, 0, 0, 5, 0, 0, 0, 5, ...} MATHEMATICAL CONCEPTS: Definition of a sequence and its terms. Meaning of convergence and divergence. How to interpret a sequence graph. Why increasingly frequent zeros do not guarantee convergence. Understanding subsequences and their limits. Applying the subsequence theorem. KEY MATHEMATICAL RULE: If a sequence converges to a limit L, every subsequence must converge to that same limit L. For this problem: • The subsequence of zeros converges to 0. • The subsequence of fives converges to 5. Because these subsequences have different limits, the original sequence cannot converge. FINAL ANSWER: lim (n → ∞) aₙ = DNE The sequence diverges. IMPORTANT REMINDER: A sequence converges only when all sufficiently late terms approach one finite number. A value appearing more frequently does not necessarily determine the limit.