( 1+4)/2= 1/2+2/ 1=2- 1/2+ 1/ 1。 The last decomposition is: 2-1/2010+1/2009, and others are similar.

Substitution; :(2- 1/2+ 1/ 1)+(2- 1/3+ 1/2)+....+(2- 1/20 10+ 1/2009)

=2*2009+2009/20 10

The second solution uses the square difference formula to subtract 2 from each term.

It can be written as = (N2+(n+1) 2)/(n * (n+1))-2+2 = 2+1/(n * (n+1)) = 2+.

The calculation method is the same as above