(negative 1 × 1 / 2) + (negative 1 / 2 × 1 / 3) + (negative 1 / 3 × 1 / 4) (negative 1 / 2012 × 1 / 2013) is equal to what?

(negative 1 × 1 / 2) + (negative 1 / 2 × 1 / 3) + (negative 1 / 3 × 1 / 4) (negative 1 / 2012 × 1 / 2013) is equal to what?

(negative 1 × 1 / 2) + (negative 1 / 2 × 1 / 3) + (negative 1 / 3 × 1 / 4) (negative 1 / 2012 × 1 / 2013)
= -（1×1/2+1/2×1/3+…… +1/2012×1/2013）
= -（1-1/2+1/2-1/3+…… +1/2012-1/2013）
= -（1-1/2013）
= -2012/2013

1 plus - 2 plus 3 plus - 4 has been added to - 2012 plus 2013

The result is one
After 2013 is calculated separately, in the previous 2012 numbers:
The sum of positive numbers is: 1 + 3 + 5 +. + 2009 + 2011 = (1 + 2011) * 2012 / 2 = 2024072
The sum of negative numbers is: - 2-4-6 -. - 2010 - 2012 = (- 2-2012) * 2012 / 2 = - 2026084
The sum of positive and negative numbers is 2024072-2026084 = - 2012
Finally, 2013-2012 = 1
Is my algorithm rather stupid? I'll consider whether there is a simpler algorithm, but I'm sure the result is 1