To determine whether the sequence $a_n = n + \frac{1}{n}$ converges, we must find its limit as $n$ approaches infinity.

Step 1: Identify the Sequence

The sequence is given by: $a_n = n + \frac{1}{n}$

Step 2: Compute the Limit

As $n$ approaches infinity, we examine the behavior of each term in the sequence: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(n + \frac{1}{n}\right)$

The term $\frac{1}{n}$ approaches 0 as $n$ becomes large. However, the term $n$ increases without bound as $n$ approaches infinity. So: $\lim_{n \to \infty} a_n = \infty$

Step 3: Conclusion

Since $\lim_{n \to \infty} a_n = \infty$, the sequence $a_n = n + \frac{1}{n}$ does not converge to a finite value. Instead, it diverges to infinity.

Would you like more details, or do you have any other questions?

Here are some related questions:

1. What is the difference between divergent and convergent sequences?
2. How can you prove that a sequence diverges to infinity?
3. Are there any sequences that oscillate rather than converge or diverge?
4. What happens to the sequence $\frac{1}{n}$ as $n$ approaches infinity?
5. How does one determine the convergence or divergence of a series?

Tip: When evaluating the convergence of a sequence, observe how each term behaves as $n$ becomes very large. If the terms increase without bound, the sequence diverges.