how to evaluate limit n tends to infinity pi/n^2(summation (r =1 to n) r sin[(pi*r/n)^2]

Introduction : In this post, we will Evaluate limit n tends to infinity pi/n^2(summation (r =1 to n) r sin[(pi*r/n)^2] by converting it into an equivalent integral that makes it very easy to evaluate.Question : Evaluate limit\[limit_{n→∞}\frac{\pi}{n^{2}}\sum_{r=1}^{n}r\sin⁡\left[\left(\frac{\pi r}{n}\right)^{2}\right]\]Solution:we know the…

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