化简:x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2006)(x+2007)分之1

问题描述:

化简:x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2006)(x+2007)分之1

1/x(x+1)=1/x-1/(x+1)
1/(x+1)(x+2)=1/(x+1)-1/(x+2)
.
1/(x+2006)(x+2007)=1/(x+2006)-1/(x+2007)
x(x+1)分之1+(x+1)(x+2)分之1+(x+2)(x+3)分之1+.+(x+2006)(x+2007)分之1
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+.+1/(x+2006)-1/(x+2007)
=1/x-1/(x+2007)
=2007/x(x+2007)