Calculus II – Section 8.1 - Q#14: Sequences | Exponential Limits

CALCULUS II | SECTION 8.1 — SEQUENCES Question 14: Exponential Limits In this lesson, we determine whether an exponential sequence converges or diverges. We explain the mathematical concepts, formulas, and calculations step by step, using graphs and numerical examples to verify the result. PROBLEM: Determine whether the following sequence converges or diverges. If it converges, find its limit. aₙ = (1 + 4/n)ⁿ WHAT YOU WILL LEARN: 1. The definition of a sequence and convergence. 2. How to recognize an exponential limit. 3. Why 1^∞ is an indeterminate form. 4. How to apply the exponential limit formula. 5. How to derive the result using the definition of e. 6. How to verify convergence using numerical values and graphs. 7. Common mistakes and strategies for calculus exams. KEY FORMULA: lim (n → ∞) (1 + a/n)ⁿ = eᵃ FINAL ANSWER: lim (n → ∞) (1 + 4/n)ⁿ = e⁴ e⁴ ≈ 54.59815 Conclusion: The sequence converges to e⁴.