The given sequence is:

$a_n = \left(1 + \frac{1}{n}\right)^n$

This sequence is a well-known limit that defines the number e (Euler's number). We can analyze the behavior of this sequence as $n \to \infty$ to determine if it converges.

Convergence Analysis:

As $n \to \infty$, the sequence approaches the mathematical constant e, which is approximately 2.71828. This result comes from a fundamental limit in calculus:

$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$

Thus, the sequence $a_n$ converges to e as $n$ increases.

Conclusion:

The sequence is convergent and converges to the value $e$.

Would you like to explore further details on how this limit is derived or ask any other questions?

Here are five related questions that can help you deepen your understanding:

1. How can we use the definition of limits to prove the convergence of this sequence?
2. What is the significance of Euler's number $e$ in mathematics?
3. How does this sequence relate to the exponential function?
4. What is the rate of convergence of the sequence as $n$ increases?
5. Can this type of limit be generalized for other sequences?

Tip: Understanding how limits work is crucial in calculus. Studying different types of limits, like this one that defines $e$, can strengthen your grasp of growth rates and convergence in sequences and series.