In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.
aₙ = 2ⁿ 3ⁿ / n!

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First, write down the given sequence explicitly: $a_n = \frac{2^n 3^n}{n!} = \frac{(2 \cdot 3)^n}{n!} = \frac{6^n}{n!}$.

To determine if the sequence is monotonic, consider the ratio $\frac{a_{n+1}}{a_n}$ and analyze whether it is greater than, less than, or equal to 1. Compute this ratio as $\frac{a_{n+1}}{a_n} = \frac{6^{n+1} / (n+1)!}{6^n / n!} = \frac{6^{n+1}}{6^n} \cdot \frac{n!}{(n+1)!} = \frac{6}{n+1}$.

Analyze the ratio $\frac{6}{n+1}$: for values of $n$ where $\frac{6}{n+1} > 1$, the sequence is increasing; where $\frac{6}{n+1} < 1$, the sequence is decreasing. This helps identify intervals of monotonicity.

To check if the sequence is bounded, consider the behavior of $a_n = \frac{6^n}{n!}$ as $n$ becomes very large. Recall that factorial growth ($n!$) eventually outpaces exponential growth (\$6^n$), which suggests the sequence might approach zero for large $n$.

Summarize the findings: use the ratio test to determine monotonicity intervals and the growth comparison between \$6^n$ and $n!$ to conclude about boundedness.

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Monotonic Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing. To determine monotonicity, compare consecutive terms (aₙ and aₙ₊₁) to see if the sequence consistently increases or decreases. This helps understand the behavior and trend of the sequence over its domain.

Bounded Sequences

A sequence is bounded if there exists a real number M such that all terms of the sequence lie within the interval [-M, M]. Checking boundedness involves finding upper and lower limits that the sequence does not exceed, which is crucial for understanding convergence and stability.

Factorials and Growth Rates

Factorials (n!) grow faster than exponential functions like 2ⁿ or 3ⁿ as n becomes large. Understanding the relative growth rates of factorials versus exponentials is essential to analyze the behavior of the sequence aₙ = (2ⁿ 3ⁿ) / n!, especially when determining limits, monotonicity, and boundedness.

In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.aₙ = 2ⁿ 3ⁿ / n!