lim1/n[1^2 (1 1/n)^2 (1 2/n)^2 .......(1 (n 1)/n)^2]求极限 不要用微积分来解，请用最基本的方法来解答

Question

lim1/n[1^2+(1+1/n)^2+(1+2/n)^2+.......(1+(n+1)/n)^2]

Answer 1

很容易，展开就行，如下：(这个图片你在这看可能看不清楚，拖动一下，就会打开分辨率很高的图片)

解题时用到了前n项和公式

Answer 2

(1+k/n)²=1+k*2/n+k²/n²
所以1²+(1+1/n)²+(1+2/n)²+.......(1+(n+1)/n)²
=n+2+(n+2)(n+1)/(2n)+(1²+2²+……+n²+(n+1)²)/n²
lim [1²+(1+1/n)²+(1+2/n)²+.......(1+(n+1)/n)²]/n
=lim [n+2+(n+2)(n+1)/(2n)+(1²+2²+……+n²+(n+1)²)/n²]/n
=lim [1+2/n+(n+2)(n+1)/(2n²)+(1²+2²+……+n²+(n+1)²)/n^3]
=1+0+1/2+lim (1²+2²+……+n²+(n+1)²)/n^3
平方和公式: 1²+2²+……+n²+(n+1)²=(n+1)(n+2)(2n+3)/6
所以原极限=1+0+1/2+2/6=11/6本回答被网友采纳

Answer 3

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