Problema: Encontrar el siguiente límite $${\displaystyle \lim_{n\rightarrow\infty}\left[\left(1+\frac{1}{n}\right)\sin\frac{\pi}{n^{2}}+\left(1+\frac{2}{n}\right)\sin\frac{2\pi}{n^{2}}+\ldots+\left(1+\frac{n-1}{n}\right)\sin\frac{\left(n-1\right)}{n^{2}}\pi\right]}$$.
Intento: Observar que \begin{align*} & \lim_{n\rightarrow\infty}\left[\left(1+\frac{1}{n}\right)\sin\frac{\pi}{n^{2}}+\left(1+\frac{2}{n}\right)\sin\frac{2\pi}{n^{2}}+\ldots+\left(1+\frac{n-1}{n}\right)\sin\frac{\left(n-1\right)}{n^{2}}\pi\right]\\ = & \lim_{n\rightarrow\infty}\left[\left(1+\frac{1}{n}\right)\left(\frac{\pi}{n^{2}}+O\left(\frac{\pi^{3}}{n^{6}}\right)\right)+\ldots+\left(1+\frac{n-1}{n}\right)\left(\left(\frac{n-1}{n^{2}}\right)\pi+O\left(\frac{\left(n-1\right)^{3}\pi^{3}}{n^{6}}\right)\right)\right]\\ = & \lim_{n\rightarrow\infty}\left[\left(\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)\frac{k\pi}{n^{2}}\right)+\left(\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)O\left(\frac{k^{3}\pi^{3}}{n^{6}}\right)\right)\right]\\ = & \lim_{n\rightarrow\infty}\left[\left(\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)\frac{k\pi}{n^{2}}\right)+\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)O\left(\frac{1}{n^{3}}\right)\right]\tag{1} \end{align*} Tenga en cuenta que \begin{align*} \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)\frac{1}{n^{3}} & =\lim_{n\rightarrow\infty}\left(\frac{1}{n^{2}}+\sum_{k=1}^{n}\frac{k}{n^{4}}\right)\leq\lim_{k\rightarrow\infty}\left(\frac{1}{n^{2}}+\frac{1}{n^{3}}\right)=0. \end{align*} Por lo tanto \begin{align*} \left(1\right) & =\lim_{n\rightarrow\infty}\left[\sum_{k=1}^{n}\left(1+\frac{k}{n}\right)\frac{k\pi}{n^{2}}\right]=\pi\lim_{n\rightarrow\infty}\left[\sum_{k=1}^{n}\left(\frac{k}{n}+\left(\frac{k}{n}\right)^{2}\right)\frac{1}{n}\right]\\ & =\pi\int_{0}^{1}x+x^{2}dx=\frac{5\pi}{6}. \end{align*}
Pregunta no sé si lo hice bien.
