求\(\displaystyle \lim_{n\to\infty}\frac{\left(\frac{1}{2n}\right)^p+\left(\frac{2}{2n}\right)^p+\ldots+\left(\frac{2n}{2n}\right)^p}
{\left(\frac{1}{2}+\frac{1}{2n}\right)^p+\left(\frac{1}{2}+\frac{2}{2n}\right)^p+\ldots+\left(\frac{1}{2}+\frac{n}{2n}\right)^p}\)之值\((p>0)\)
。
[解答]
切片切的順手就好的,還是喜歡把範圍限在0到1
原式\( \displaystyle=2\lim_{n\to\infty}\frac{\displaystyle\frac{1}{2n}\sum\limits_{k=1}^{2n}\left(\frac{k}{2n}\right)^p}{\displaystyle\frac{1}{n}\sum\limits_{k=1}^{n}\left(\frac{1}{2}+\frac{k}{2n}\right)^p}=2\times\frac{\displaystyle\int_0^1x^pdx}{\displaystyle\int_0^1\left(\frac{1}{2}+\frac{x}{2}\right)^pdx}=\frac{2^{p+1}}{2^{p+1}-1} \)
111.2.14補充
105鳳山高中,
https://math.pro/db/viewthread.php?tid=2511&page=2#pid15490