利用定积分定义求解lim(n→∞){n*[1/(n+1)^2+1/(n+2)^2+…1/(n+n)^2]}

题目

利用定积分定义求解lim(n→∞){n*[1/(n+1)^2+1/(n+2)^2+…1/(n+n)^2]}

答案

lim[n→+∞] n * [1/(n + 1)² + 1/(n + 2)² + ...+ 1/(n + n)²]
= lim[n→+∞] n * {1/[n(1 + 1/n)]² + 1/[n(1 + 2/n)]² + ...+ 1/[n(1 + n/n)]²}
= lim[n→+∞] n * (1/n²)[1/(1 + 1/n)² + 1/(1 + 2/n)² + ...+ 1/(1 + n/n)²]
= lim[n→+∞] 1/n * [1/(1 + 1/n)² + 1/(1 + 2/n)² + ...+ 1/(1 + n/n)²]
= lim[n→+∞] 1/n * Σ(k=1→n) 1/(1 + k/n)²
= lim[n→+∞] (2 - 1)/n * Σ(k=1→n) 1/[1 + k(2 - 1)/n]²
= ∫[1→2] 1/x² dx
= - 1/x |[1→2]
= - (1/2 - 1)
= 1/2
这里的Δx = (2 - 1)/n = 1/n
区间是1 + 1/n,1 + 2/n,1 + 3/n,...,1 + k/n,...,1 + n/n