SOLUTION: Could you please teach me how to evaluate the sum of the following series? 1 + 2 – 3 + 4 – 5 + 6 – 7 + … + 2000

1 + 2 – 3 + 4 – 5 + 6 – 7 + … + 2000

That has 2000 terms.  Let's write in the next to
the last three terms:

1 + 2 – 3 + 4 – 5 + 6 – 7 + … + 1998 - 1999 + 2000

We put parentheses in like this:

1 + (2 – 3) + (4 – 5) + (6 – 7) + … + (1998 - 1999) + 2000 

Only the 1 on the left and the 2000 on the right are not in
parentheses.  Those are the only two numbers that are not
in parentheses so the other 1998 of the 2000 numbers are in
parentheses.  Since there are two numbers in each set of 
parentheses there are one-half of 1998 or 999 sets of 
parentheses.

Every one of those 999 sets of parentheses has -1 in it.

So we have 999 negative one's which makes -999 plus the 1 on the left 
and the 2000 on the right.  That makes the sum -999+1+2000 = 1002

Edwin