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この問題について途中式を含めて、徐々に解を導き出していこう。

The given expression is:

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

### Step-by-step solution:

1. **Rewriting the sum**:  
   For each fixed \( n \), the sum is given by:

   \[
   S_n = \sum_{k=1}^{n} \frac{1}{n^2 + k}
   \]
   
   We want to evaluate the limit of this sum as \( n \to \infty \).

2. **Approximation for large \( n \)**:  
   When \( n \) is large, for each \( k \), we can approximate the denominator \( n^2 + k \) by \( n^2 \) since \( k \) is much smaller than \( n^2 \). Thus,

   \[
   \frac{1}{n^2 + k} \approx \frac{1}{n^2}
   \]

   This approximation allows us to simplify the sum:

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

   As \( n \to \infty \), \( \frac{1}{n} \to 0 \). This suggests that the limit of the sum might approach 0.

3. **Improving the approximation**:  
   To get a more precise result, observe that the sum resembles a Riemann sum. We rewrite the sum in terms of an integral approximation. Notice that:

   \[
   S_n = \sum_{k=1}^{n} \frac{1}{n^2} \cdot \frac{1}{1 + \frac{k}{n^2}} \approx \frac{1}{n} \sum_{k=1}^{n} \frac{1}{1 + \frac{k}{n^2}}
   \]

   As \( n \to \infty \), this sum behaves like the Riemann sum for the integral of the function \( f(x) = \frac{1}{x^2} \) over an interval.

   Thus, the sum tends to zero, leading to the result:

   \[
   \lim_{n \to \infty} S_n = 0
   \]

### Final Answer:
\[
\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n^2 + k} = 0
\]

---

Would you like further clarification or more details?

Here are some related questions to explore:
1. How do Riemann sums help approximate sums like this?
2. What if the denominator had a different dependence on \( n \) (e.g., \( n^3 + k \))?
3. How would the result change if the summation limit \( n \) was replaced with another variable?
4. Can we apply the same method to a more complex summation involving higher powers of \( n \)?
5. What is the physical or geometric interpretation of such limits?

**Tip:** Recognizing when a sum can be approximated by an integral is a powerful technique in calculus, especially for large \( n \).

Explore the solution to the summation lim_{n -> ∞} ∑_{k=1}^n 1 / (n^2 + k) using approximation techniques like Riemann sums. Detailed steps and mathematical reasoning are provided. Suitable for college-level calculus students.

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