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证1/1*2*3+1/2*3*4+...+1/N(N+1)(N+2)＜4
题目详情
证1/1*2*3 +1/2*3*4+...+1/N(N+1)(N+2)＜4
▼优质解答
答案和解析
1/N(N+1)(N+2)=(1/n(n+1)-1/(n+1)(n+2))*1/2
所以,
1/1*2*3 +1/2*3*4+...+1/N(N+1)(N+2)
=[(1/1*2-1/2*3)+(1/2*3-1/3*4)+...+(1/n(n+1)-1/(n+1)(n+2)]*1/2
=(1/2-1/(n+1)(n+2))*1/2
<1/2*1/2=1/4
所以,
1/1*2*3 +1/2*3*4+...+1/N(N+1)(N+2)＜1/4
所以,
1/1*2*3 +1/2*3*4+...+1/N(N+1)(N+2)
=[(1/1*2-1/2*3)+(1/2*3-1/3*4)+...+(1/n(n+1)-1/(n+1)(n+2)]*1/2
=(1/2-1/(n+1)(n+2))*1/2
<1/2*1/2=1/4
所以,
1/1*2*3 +1/2*3*4+...+1/N(N+1)(N+2)＜1/4
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