x=2,y=1,1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+…+1/(x+2010)(y+2010)=?
问题描述：
x=2,y=1,1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+…+1/(x+2010)(y+2010)=?
答
1/xy+1/(x+1)(y+1)+1/(x+2)(y+2)+…+1/(x+2010)(y+2010)
=1/1*2+1/2*3+1/3*4+......../2011*2012
=(2-1)/1*2+(3-2)/2*3+(4-3)/3*4+.....(2012-2011)/2011*2012)
=1-1/2+1/2-1/3+1/3+.....-2011/2012
=1/2012
答
1=x-y
1/(x+n)(y+n)=[(x+n)-(y+n)]/(x+n)(y+n)=1/(y+n)-1/(x+n)
原式=(1/1-1/2)+(1/2-1/3)+...+(1/2011-1/2012)=1-1/2012=2011/2012
答
先说明一个公式：1/n(n+1)=1/n-1/(n+1)
这个公式通分就可以证明
把x,y代进去
1/1*2+1/2*3+1/3*4+…………+1/2011*2012
=1/1-1/2+1/2-1/3+1/3-1/4+…………+1/2011-1/2012
=2011/2012