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The problem provided defines a sequence \( a_n \) as follows:

\[
a_n = \frac{n^n}{1 + 2^2 + 3^3 + \dots + n^n}
\]

The task might involve evaluating the behavior of \( a_n \) as \( n \) grows larger or simplifying this expression if possible.

Would you like to:
1. Determine the limit of \( a_n \) as \( n \to \infty \)?
2. Calculate specific terms of the sequence (e.g., \( a_1, a_2, \) etc.)?
3. Analyze the growth of the denominator in comparison to \( n^n \)?
4. Explore whether the sequence \( a_n \) converges or diverges?
5. Determine an approximation for large \( n \)?

Let me know if you want a detailed solution or have any questions! 

**Tip:** When dealing with sequences and series, analyzing the terms’ growth rate (e.g., polynomial vs. exponential) often helps in understanding convergence or divergence.

Determine the behavior of the sequence \( a_n = \frac{n^n}{1 + 2^2 + 3^3 + \dots + n^n} \) as \( n \) grows larger or simplify the expression if possible.

Explore the sequence \( a_n = \frac{n^n}{1 + 2^2 + 3^3 + \dots + n^n} \) and understand its behavior as \( n \) approaches infinity. This problem delves into exponential growth, convergence, and sequence analysis suitable for college-level students.

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