This problem involves evaluating the limit of a sum as $n \to \infty$. The given expression is:

$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n^2 + k}$

We can approach this sum by interpreting it as a Riemann sum in the limit. Let’s break it down step by step:

Step 1: Express the sum in a more manageable form

We can rewrite the sum as:

$S_n = \sum_{k=1}^{n} \frac{1}{n^2 + k}$

This sum resembles a Riemann sum when $n$ becomes large. To see this, divide both the numerator and denominator by $n^2$:

$S_n = \sum_{k=1}^{n} \frac{1/n^2}{1 + \frac{k}{n^2}}$

However, simplifying this further directly into a Riemann sum involves noting how the summand behaves as $n \to \infty$.

Step 2: Approximate the summand for large $n$

When $n$ becomes very large, we notice that $k$ remains much smaller than $n^2$. This allows us to approximate the summand as:

$\frac{1}{n^2 + k} \approx \frac{1}{n^2}$

Thus, the sum becomes approximately:

$S_n \approx \sum_{k=1}^{n} \frac{1}{n^2} = \frac{n}{n^2} = \frac{1}{n}$

Step 3: Take the limit as $n \to \infty$

Now, take the limit as $n \to \infty$:

$\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{1}{n} = 0$

Thus, the limit of the sum is:

$\boxed{0}$

The sum converges to zero as $n \to \infty$.

Would you like more details on how we approximated the sum or any other part of the process?

Here are five follow-up questions you may find helpful:

1. How does this problem relate to the concept of a Riemann sum?
2. What happens if we modify the summand by changing the powers of $n$ in the denominator?
3. Can we approach this problem using integrals?
4. Why does the approximation $\frac{1}{n^2 + k} \approx \frac{1}{n^2}$ work for large $n$?
5. How would the result change if the limit were taken for $n \to 0$?

Tip: When analyzing limits of sums, try to recognize patterns that resemble integral approximations, as they often provide deeper insights into the behavior of the function.