Question

计算1/x(x+1)+1/（x+2）（x+1）.+1/(x+2008)(x+2009)并求当x=1时,该代数式的值

Answer 1

1/x(x+1)=1/x-1/(1+x)

1/（x+2)(x+1)=1/(x+1)-1/(x+2)

.

1/(x+2008)(x+2009)=1/(x+2008)-1/(1+2009)

1/x(x+1)+1/（x+2）（x+1）.+1/(x+2008)(x+2009)=1/x-1/(1+x)+1/(x+1)-1/(x+2)+.+1/(x+2008)-1/(x+2009)

=1-1/(x+2009)

1-1/(1+2009)=1-1/2010=2009/2010