12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!

Verified step by step guidance

1. Identify the sequence given: \(a_n = \frac{8^n}{n!}\), where \(n!\) denotes the factorial of \(n\).
2. Recall that factorial \(n!\) grows much faster than any exponential function \$8^n\( as \)n$ becomes very large.
3. To analyze the limit \(\lim_{n \to \infty} \frac{8^n}{n!}\), consider the behavior of the terms in the denominator compared to the numerator as \(n\) increases.
4. Use the ratio test for sequences by examining the ratio \(\frac{a_{n+1}}{a_n} = \frac{8^{n+1}/(n+1)!}{8^n/n!} = \frac{8}{n+1}\), which approaches 0 as \(n \to \infty\).
5. Since the ratio approaches 0, the terms \(a_n\) get smaller and smaller, indicating that the limit of the sequence is 0.

Key Concepts

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Factorials and Growth Rates

Factorials (n!) grow much faster than exponential functions like 8ⁿ as n increases. Understanding this difference in growth rates helps determine the behavior of sequences involving factorials and exponentials.

Ratio Test for Sequences

The ratio test compares the ratio of consecutive terms aₙ₊₁ / aₙ to analyze the limit of a sequence. If this ratio approaches a value less than 1, the sequence tends to zero, aiding in evaluating limits involving factorials and exponentials.