\begin{equation}\text { Question: The value of } \lim _(n \rightarrow \infty) (1)/(n) \sum_(r=0)^(2 n-1) (n^(2))/(n^(2)+4 r^(2)) \text { is }\end{equation}

Question

`\begin{equation}\text { Question: The value of } \lim _(n \rightarrow \infty) (1)/(n) \sum_(r=0)^(2 n-1) (n^(2))/(n^(2)+4 r^(2)) \text { is }\end{equation}`

Answer 1

You have something resembling a Riemann sum. Multiply through the summand by 1/n², then you can write

`\displaystyle \lim_(n\to\infty) \frac1n \sum_(r=0)^(2n-1) (1)/(1+4\left(\frac rn\right)^2) = \int_0^2 (dx)/(1+4x^2) = \boxed{\frac12 \tan^(-1)(4)}`