Help me to solve! 1/2013 - 2/2013 + 3/2013 - 4/2013.....2012 /2013

what is value of x?

The given expr. = (1/2013){1 - 2 + 3 - 4 ------- - 2012}
= (1/2013){(1 + 3 + 5 + ------ + 2011) - (2 + 4 + 6 + ---- + 2012)}
= (1/2013){(1006/2)(1+ 2011) - (1006/2)(2 + 2012)}
= (503/2013){2012 - 2014}
= - 2*(503/2013)
= - 1006/2013
= - 0.49975......
Explanation : Total No. of terms = n = 2012 ,
divided into 2 groups of 1006 each,

1/2013 - 2/2013 + 3/2013 - 4/2013.....2012 /2013
= (1/2013 + 3/2013 +.....+ 2012 /2013) - (2/2013 + 4/2013 +....+ 2011/2013)
= (1 + 3 + 5 + .... + 2012)/2013 - (2 + 4 + 6 + ... + 2011)/2013
= [Summation (2n - 1)(n from 1 to 1006) - Summation (2n)(n from 1 to 1006)]/2013
= [1012036 - 1013042]/2013
= -1006/2013
= -0.4997516145...

n1=2012/2=1006
n2=2012/2=1006

solution::::
=(n1*n1-n2*(n2+1))/2013
=(1006*1006-1006*1007)/2013
=-1006/2013
=0.4997516

-.002