设M=1/(1+根号2)+1/(根号2+根号3)+……+1/(根号2007+根号2008),N=1-2+3-4+……+2007-2008,则N/(M+1)^2=?

问题描述:

设M=1/(1+根号2)+1/(根号2+根号3)+……+1/(根号2007+根号2008),N=1-2+3-4+……+2007-2008,则N/(M+1)^2=?

M=√2-1+√3-√2+...+√2007-√2006+√2008-√2007=√2008-1
N=(1-2)+(3-4)+...+(2007-2008)=-1004
N/(M+1)²=-1004/2008=-1/2

做此题须知:1/(1+√2)=(√2-1)/[(√2+1)(√2-1)]=√2-1同理可知1/(√2+√3)=√3-√2……1/(√2007+√2008)=√2008-√2007所以M=√2008-√2007+√2007-√2006……+√3-√2+√2-1=√2008-1所以(M+1)^2=2008而N=(1-2...