请帮忙计算出下面的奥数题!

∵ (1/n)÷[(1+1/2)(1+1/3)(1+1/4)×.×(1+1/n)]
=(1/n)÷{(3/2)(4/3)(5/4)×.×[(n+1)/n]}
=(1/n)÷(n+1)/2
=2/[n(n+1)]
=2[1/n-1/(n+1)] (n≥2)
∴1/2/(1+1/2) + 1/3/(1+1/2)(1+1/3) + 1/4/(1+1/2)(1+1/3)(1+1/4) + …
… + 1/2008/(1+1/2)(1+1/3）……(1+1/2008)
=2(1/2-1/3)+2(1/3-1/4)+2(1/4-1/5)+.+2(1/2008-1/2009)
=2(1/2-1/3+1/3-1/4+1/4-1/5+.+1/2008-1/2009)
=2(1/2-1/2009)
=1-2/2009
=2007/2009