Question

%5Crightarrow%20%5Cinfty%7D%20%5Cfrac%7B1%7D%7Bn%7D%20%5Csum_%7Br%3D0%7D%5E%7B2%20n-1%7D%20%5Cfrac%7Bn%5E%7B2%7D%7D%7Bn%5E%7B2%7D%2B4%20r%5E%7B2%7D%7D%20%5Ctext%20%7B%20is%20%7D%5Cend%7Bequation%7D" id="TexFormula1" title="\begin{equation}\text { Question: The value of } \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^{2}}{n^{2}+4 r^{2}} \text { is }\end{equation}" alt="\begin{equation}\text { Question: The value of } \lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^{2}}{n^{2}+4 r^{2}} \text { is }\end{equation}" align="absmiddle" class="latex-formula">

Answer 1

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

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