引用:
原帖由 may 於 2010-6-21 10:29 PM 發表 
想請教第18題，感謝。
第 18 題：
題目：求 \(\displaystyle \lim_{n\to\infty} \frac{\left(1^2+2^2+\cdots+n^2\right)\left(^5+2^5+\cdots+n^5\right)}{\left(1^3+2^3+\cdots+n^3\right)\left(1^4+2^4+\cdots+n^4\right)}\) 之值。
解答：
\(\displaystyle 1+2+\cdots+n=\frac{1}{2}n^2+O(n)\)
\(\displaystyle 1^2+2^2+\cdots+n^2=\frac{1}{3}n^3+O(n^2)\)
\(\displaystyle 1^3+2^3+\cdots+n^3=\frac{1}{4}n^4+O(n^3)\)
\(\displaystyle 1^4+2^4+\cdots+n^4=\frac{1}{5}n^5+O(n^4)\)
\(\displaystyle 1^5+2^5+\cdots+n^5=\frac{1}{6}n^6+O(n^5)\)
所求 \(\displaystyle =\lim_{n\to\infty}\frac{\displaystyle\frac{n^3}{3}\cdot\frac{n^6}{6}+O(n^7)}{\displaystyle\frac{n^4}{4}\cdot\frac{n^5}{5}+O(n^7)}=\frac{10}{9}.\)
