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题:计算\(\,\displaystyle\lim_{n \to \infty}{\small\frac{\ln{n}}{n^2}}\int_{1}^{n} \lfloor x\rfloor^{x-\lfloor x\rfloor} dx\)
解:\(\,\displaystyle\int_{k}^{k+1} \lfloor x\rfloor^{x-\lfloor x\rfloor} dx=k^{-k}\int_k^{k+1}k^xdx=\small\frac{k-1}{\ln k}\)
\(\therefore\quad\)所求极限\(\overset{\text{Stolz}}{=}\displaystyle\lim_{n\to\infty}\frac{\int_n^{n+1}\lfloor x\rfloor^{x-\lfloor x\rfloor}dx}{\frac{(n+1)^2}{\ln(n+1)}-\frac{n^2}{\ln n}}=\lim_{n\to\infty}\frac{\frac{n-1}{\ln n}}{\frac{(n+1)^2}{\ln(n+1)}-\frac{n^2}{\ln n}}=\small\frac{1}{2}\)
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