Step 1: Understanding the Concept:
This problem explores the concept of the "Limit of a Sum as a Definite Integral," which is based on the Riemann sum definition of integration.
When we have a sum of terms where each term varies with an index \(r\) and a parameter \(n\), and \(n\) approaches infinity, the sum can often be converted into an integral over the interval \([0, 1]\).
The process involves factoring out a term \(\frac{1}{n}\) to act as the differential \(dx\) and identifying a function \(f(r/n)\).
This transformation allows us to use the powerful tools of integral calculus to evaluate discrete sums that would otherwise be impossible to calculate term by term.
It essentially treats the sum as an area under a curve composed of infinitely many rectangles of width \(1/n\).
Step 2: Key Formula or Approach:
1. Riemann Sum Transformation: \(\lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{n} f(\frac{r}{n}) = \int_{0}^{1} f(x) dx\).
2. Standard Integral: \(\int \frac{x}{x^2+1} dx = \frac{1}{2} \log(x^2+1)\).
Step 3: Detailed Explanation:
Let the limit be \(L = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{r}{r^2+n^2}\).
To put this into the Riemann form, we divide the numerator and denominator of the general term by \(n^2\):
\[ \frac{r}{r^2+n^2} = \frac{r/n^2}{(r^2/n^2) + 1} = \frac{1}{n} \cdot \frac{r/n}{(r/n)^2 + 1} \]
Substituting this back into the summation:
\[ L = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{n} \cdot \frac{r/n}{(r/n)^2 + 1} \]
Now, we apply the transformations: \(\frac{r}{n} \to x\) and \(\frac{1}{n} \to dx\). The limits are from \(r=1\) (\(1/n \to 0\)) to \(r=n\) (\(n/n \to 1\)).
\[ L = \int_{0}^{1} \frac{x}{x^2+1} dx \]
To evaluate this integral, use the substitution \(u = x^2+1\), which gives \(du = 2x dx\).
\[ L = \frac{1}{2} \int_{1}^{2} \frac{1}{u} du = \frac{1}{2} [ \log |u| ]_{1}^{2} \]
\[ L = \frac{1}{2} (\log 2 - \log 1) = \frac{1}{2} \log 2 \]
Step 4: Final Answer:
The limit converges to the value \(\frac{1}{2} \log 2\).