定积分定义

\[ \begin{aligned} I &= \lim_{ n \to \infty } \sum_{k=1}^{n} \frac{k^{2025}}{n^{2026}}\int_{k}^{n} \frac{1}{1+\left( \frac{x}{n} \right)^{2027}} \mathrm{~d} \left( \frac{x}{n} \right) \\ &= \lim_{ n \to \infty } \frac{1}{n}\sum_{k=1}^{n} \left( \frac{k}{n} \right)^{2025}\int_{\frac{k}{n}}^{1} \frac{1}{1+y^{2027}} \mathrm{~d} y\\ &= \int_{0}^{1} \left[ x^{2025}\cdot \int_{x}^{1} \frac{\mathrm{~d} y }{1+y^{2027}} \mathrm{~d} y \right] \mathrm{~d} x \\ &\xlongequal{交换积分次序} \int_{0}^{1} \frac{1}{1+y^{2027}} \mathrm{~d} y \int_{0}^{y} x^{2025} \mathrm{~d} x \\ &= \frac{1}{2026}\int_{0}^{1} \frac{y^{2026}}{1+y^{2027}} \mathrm{~d} y \\ &= \frac{1}{2026}\cdot \frac{1}{2027}\int_{0}^{1} \frac{1}{1+y^{2027}} \mathrm{~d} (y^{2027})\\ &= \frac{1}{2026}\cdot \frac{1}{2027} \cdot \ln(1+y^{2027})\bigg|_{0}^1\\ &= \frac{1}{2026}\cdot \frac{1}{2027}\cdot\ln 2 \end{aligned} \]