The given sequence $\{a_n\}$ is defined as:

$a_n = \frac{n}{n^2 + 1}$

This represents a sequence where each term is computed based on the index $n$, with the general formula involving both $n$ and its square.

Properties and Analysis:

1. Limit as $n \to \infty$:
   As $n$ increases, the value of the sequence approaches a limit. We can analyze the limit of $a_n$ as $n \to \infty$: $\lim_{n \to \infty} \frac{n}{n^2 + 1} = \lim_{n \to \infty} \frac{n}{n^2(1 + \frac{1}{n^2})} = \lim_{n \to \infty} \frac{1}{n(1 + \frac{1}{n^2})} = 0$ Therefore, the sequence tends towards zero as $n$ becomes large.

2. Behavior for small $n$:
   For small values of $n$, we can calculate specific terms: $a_1 = \frac{1}{1^2 + 1} = \frac{1}{2}, \quad a_2 = \frac{2}{2^2 + 1} = \frac{2}{5}, \quad a_3 = \frac{3}{3^2 + 1} = \frac{3}{10}$ So the terms decrease as $n$ increases.

3. Monotonicity and boundedness:
   The sequence is decreasing for $n \geq 1$ since $a_{n+1} < a_n$ for positive $n$, and all terms are positive. It is also bounded below by 0, as seen in the limit behavior.

Would you like further details on these concepts or on how the sequence behaves for larger values of $n$?

Here are 5 related questions for deeper exploration:

1. How do you prove rigorously that the sequence is decreasing?
2. Can the sequence $a_n$ be expressed as a sum or product of simpler sequences?
3. What happens if you modify the denominator, for example $n^2 - 1$?
4. How would the sequence change if the numerator was a constant instead of $n$?
5. How does the rate of convergence to zero compare with other similar sequences?

Tip: Investigating the derivative of the general term can help confirm whether a sequence is increasing or decreasing.