(1 / 2 + 1 / 3 +... 1 / 2005) times (1-1 / 2 + 1 / 3 +... 1 / 2004) - (1 + 1 / 2 +... 1 / 2005) times (1 / 2 + 1 / 3 +... 1 / 2004)

(1 / 2 + 1 / 3 +... 1 / 2005) times (1-1 / 2 + 1 / 3 +... 1 / 2004) - (1 + 1 / 2 +... 1 / 2005) times (1 / 2 + 1 / 3 +... 1 / 2004)

Let a = 1 / 2 + 1 / 3 +... 1 / 2004
Original formula = (a + 1 / 2005) (1 + a) - (1 + A + 1 / 2005) a
=a(1+a)+1/2005(1+a)-a(1+a)-1/2005a
=1/2005(1+a)-1/2005a
=1/2005+1/2005a-1/2005a
=1/2005