Find the value of $\lim_{n\to \infty}\sum_{i=0}^n\frac{n}{n^2+i^2}$ I can't solve this problem and I've seen that some books like Spivak or Apostol use integrals to solve it but my teacher said that we can't integrate, I would appreciate your help :)
2026-04-02 15:45:46.1775144746
Find the value of $\lim_{n\to \infty}\sum_{i=0}^n\frac{n}{n^2+i^2}$
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$$\sum_{i=0}^n\frac n{n^2+i^2}=\frac1n\sum_{i=0}^n\frac1{1+\left(\frac in\right)^2}\xrightarrow[n\to\infty]{}\int\limits_0^1\frac1{1+x^2}dx=\ldots$$