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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

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.

lim_{n -> ∞} sum_{k=1}^{n} (1 / (n^2 + k))

Solve the limit of the summation problem lim_{n -> ∞} sum_{k=1}^{n} (1 / (n^2 + k)) using a Riemann sum approximation. Learn how to break down and approximate infinite sums in this detailed step-by-step solution. Ideal for students learning calculus concepts such as limits and summations.

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