https://file.aiquickdraw.com/mathbot/file/1726999945347pezrle1a.jpg

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.

Understand how the sequence (1 + 1/n)^n converges to Euler's number e as n approaches infinity. This calculus problem highlights key concepts in limits and sequences, perfect for students in Grades 11-12.

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