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.

Learn how to determine whether the sequence n+1/n converges with a detailed step-by-step solution. Understand the concept of sequences, limits, and convergence vs. divergence in mathematics.

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