Evaluate the limit as n approaches infinity of sum from $k=1$ to n of n / ($n^2 + k^2$).

Step 1: Rewrite the sum in a suitable form

To convert the sum into a form that can be recognized as a definite integral, we first divide the numerator and denominator by $n^2$. This allows us to express the term inside the sum in terms of $\frac{k}{n}$.

Step 2: Simplify the expression

After dividing by $n^2$, we simplify the expression. The n in the numerator cancels with one of the n 's in the denominator, leaving $\frac{1}{n}$ outside the term 1 + ($\frac{k}{n}$)^2.

Step 3: Convert the sum to a definite integral

We recognize this limit of a sum as a definite integral. Here, $\frac{k}{n}$ is replaced by x, and $\frac{1}{n}$ is replaced by dx. The limits of integration are from $x = \lim_{n \to \infty} \frac{1}{n} = 0$ to $x = \lim_{n \to \infty} \frac{n}{n} = 1$.

Step 4: Evaluate the integral

The integral of $\frac{1}{1 + x^2}$ with respect to x is $\tan^{-1}(x)$. We now need to evaluate this antiderivative at the upper and lower limits of integration.

Step 5: Apply the limits of integration

We apply the Fundamental Theorem of Calculus by substituting the upper limit (1) and the lower limit (0) into the antiderivative and subtracting the results.

Step 6: Calculate the final value

We know that $\tan^{-1}(1)$ $\frac{\pi}{4}$ (since $\tan$($\frac{\pi}{4}$) = 1) and $\tan^{-1}(0) = 0 (since \tan(0) = 0)$. Subtracting these values gives the final result.

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