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

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Identify the sequence given: $a_n = \frac{8^n}{n!}$, where $n!$ denotes the factorial of $n$.

Recall that factorial $n!$ grows much faster than any exponential function \$8^n$ as $n$ becomes very large.

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.

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 $n \to \infty$.

Since the ratio approaches 0, the terms $a_n$ get smaller and smaller, indicating that the limit of the sequence is 0.

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주요 개념

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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.

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