慕容小青
幼苗
共回答了20个问题采纳率:95% 举报
n*(n+1)+(n+1)²〉2n*(n+1) 所以1/[n*(n+1)+(n+1)²]<1/2n(n+1)=1/2 ×[1/n -1/(n+1)]
则1/(1*2+2²) + 1/(2*3+3²) +.+1/[n*(n+1)+(n+1)²]<1/2 ×[1-1/2+1/2-1/3+.+1/n-1/(n+1)]=1/2×n/(n+1)
以为n为正整数则当n=5时1/(1*2+2²) + 1/(2*3+3²) +1/[n*(n+1)+(n+1)²]<5/12成立
1年前
追问
8
eaipian
举报
应该是: 1/[n*(n+1)+(n+1)²]=1/(n+1)(2n+1) <1/(n+1)2n = 1/2[1/n- 1/(n+1)] 1/(1*2+2²)+ 1/(2*3+3²)+1/[n*(n+1)+(n+1)²]<1/(1*2+2²)+ 1/2[1/2- 1/3 +1/3-1/4+1/4-1/5...] <1/6+ (1/2)*(1/2)=5/12