Definite Integral As Limit Of Sum

A limit of a sum indexed from 1 to n with a 1/n scaling factor is the signature of the Riemann-sum definition, which lets us replace a hard discrete sum by a continuous definite integral. The strategy is to factor out 1/n as the sample spacing and identify k/n as the sample point, so we rewrite \sum_{k=1}^{n}\frac{n}{n^2+k^2} = \sum_{k=1}^{n}\frac{1}{n}\cdot\frac{1}{1 + (k/n)^2}. As n \to \infty, setting x = k/n and dx = 1/n, the sum converges to \int_0^1 \frac{dx}{1 + x^2} = [\arctan x]_0^1 = \arctan 1 - \arctan 0 = \pi/4. Option \pi/2 would arise from integrating over [0, \infty) instead of the unit interval. Option 1 ignores the arctangent structure of the integrand. Option \ln 2 corresponds to a different integrand such as 1/(1+x). This conversion embodies the deep bridge between discrete summation and continuous integration. As a final plausibility check, every term in the sum is positive and bounded above, the partial sums increase steadily toward a finite limit, and the value \pi/4 \approx 0.785 lies plausibly between the magnitudes of the first and last terms for large n, so both the analytic conversion and the numerical intuition agree on the answer.