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A sequence is monotonic if it consistently increases or decreases. To check if a sequence is monotonic, we often examine consecutive terms. For the sequence \(a_n = \frac{2^n 3^n}{n!}\), we compared the ratio of consecutive terms: \[\frac{a_{n+1}}{a_n} = \frac{6}{n+1}.\] Notice that this fraction becomes less than 1 when \( n \geq 6 \). When this happens, each term becomes smaller than the previous one, meaning the sequence decreases at least from this point onwards.

• If a sequence is continuously increasing, it is called an increasing sequence.
• If a sequence is continuously decreasing, it is called a decreasing sequence.
• Since our sequence decreases after a certain point, it acts as a decreasing sequence beyond \( n = 6 \).

A sequence is considered bounded if all its terms stay within a fixed range. For the sequence \(a_n = \frac{2^n 3^n}{n!}\), we noted this feature by observing its behavior as \( n \to \infty \). The key observation here is the role of \( n! \), the factorial in the denominator.
- The factorial grows very rapidly, outpacing exponential growth like \( 6^n \).
- As \( n \) becomes large, \( n! \) becomes significantly larger than \( 2^n 3^n \), causing \( a_n \) to shrink towards 0.
- Therefore, the sequence is bounded above, as \( a_n \leq \frac{2^n 3^n}{6^n} = 1 \) for large \( n \).
- Since \( a_n \geq 0 \) naturally, it is also bounded below.
In short, \( a_n \) is squeezed between 0 and an upper bound close to 1, keeping it nicely bounded.

Factorials like \( n! \) can significantly influence the growth or decline of a sequence. A factorial is the product of all integers up to \( n \), making the number grow remarkably fast as \( n \) grows.
For the sequence \(a_n = \frac{2^n 3^n}{n!}\):

• The numerator \(2^n 3^n\) rises exponentially, but
• The denominator \(n!\) rises even faster.

This rapid growth of \( n! \) is crucial in making \( a_n \) approach 0 as \( n \to \infty \). It's why factorials can sometimes dominate in sequences and change their behavior dramatically compared to sequences without factorials.

• Factorials are useful in determining limits and convergence of sequences.
• In this example, the factorial ensures that \( a_n \) remains bounded and approaches 0.