The expression $\frac{n+1}{n^2}$ converges because, as $n$ increases, the value of the expression approaches zero.

Here's a step-by-step explanation:

1. Break down the expression: $\frac{n+1}{n^2} = \frac{n}{n^2} + \frac{1}{n^2} = \frac{1}{n} + \frac{1}{n^2}$ This shows that the expression can be split into two simpler terms: $\frac{1}{n}$ and $\frac{1}{n^2}$.

2. Analyze the terms:

   • The term $\frac{1}{n}$ decreases as $n$ increases and approaches 0.
   • The term $\frac{1}{n^2}$ also decreases as $n$ increases, but it does so much faster than $\frac{1}{n}$ and also approaches 0.

3. Add the terms:

   • As $n$ increases, both $\frac{1}{n}$ and $\frac{1}{n^2}$ approach 0. Therefore, their sum, $\frac{1}{n} + \frac{1}{n^2}$, also approaches 0.

4. Conclusion: Since the entire expression $\frac{n+1}{n^2}$ tends toward 0 as $n$ grows larger, we say that the expression converges to 0. The convergence is because both individual components of the expression $\frac{1}{n}$ and $\frac{1}{n^2}$ converge to 0 as $n$ approaches infinity.

Do you want more details or have any questions?

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Here are five related questions you might find interesting:

1. How do you prove the convergence of a sequence formally?
2. What are other examples of sequences that converge to 0?
3. What is the difference between convergence and divergence in sequences?
4. How does the rate of convergence differ between $\frac{1}{n}$ and $\frac{1}{n^2}$?
5. Can the ratio test be used to determine the convergence of $\frac{n+1}{n^2}$?

Tip: When determining the convergence of a sequence or series, breaking down the expression into simpler parts can make it easier to analyze.