求极限 $$ex vlm{n}dfrac{(n^2+1)(n^2+2)cdots(n^2+n)}{(n^2-1)(n^2-2)cdots(n^2-n)}. eex$$
解答: 还记得对数不等式么: $$ex dfrac{x}{1+x}<ln(1+x)<x,quad x>0. eex$$ 我们有 $$eex ea lndfrac{n^2+i}{n^2-i}&=lnsex{1+dfrac{2i}{n^2-i}} <dfrac{2i}{n^2-i}leq dfrac{2i}{n^2-n},\ lndfrac{n^2+i}{n^2-i}&>dfrac{dfrac{2i}{n^2-i}}{1+dfrac{2i}{n^2-i}} =dfrac{2i}{n^2+i}geq dfrac{2i}{n^2+n}. eea eeex$$ 相加而有 $$ex 1< ln prod_{k=1}^n dfrac{n^2+i}{n^2-i}<dfrac{n^2+n}{n^2-n}. eex$$ 令 $n oinfty$ 即得原极限 $=e$.