i'm not sure if this is the way you're supposed to solve this, but i do believe i did solve it.
it doesn't look like an arithmetic series or a geometric series so those formulas don't appear to work with this.
the first term is x
the second term is 2x^2
the third term is presumably 3x^3
the fourth term is presumably 4x^4
the 2013th term is 2013x^2013 as shown.
when x = -1, this turns out to be:
first term is -1
second term is 2
third term is -3
fourth term is 4
etc.
i then looked for a pattern.
the pattern i found is shown here.
n An Sn
1 -1 -1
2 2 1
3 -3 -2
4 4 2
5 -5 -3
6 6 3
7 -7 -4
8 8 4
9 -9 -5
10 10 5
11 -11 -6
12 12 6
13 -13 -7
14 14 7
15 -15 -8
16 16 8
17 -17 -9
18 18 9
19 -19 -10
20 20 10
n is the term in the series.
An is the nth term in the series.
Sn is the sum of the n terms in the series.
if you look at the series, you will see that the sum of n terms, where n is even, is equal to exactly one half of the nth term.
the sum of 16 terms is 8
the sum of 18 terms is 9
the sum of 20 terms is 10
if you look at the sum of the odd terms, you will see that the sum of the n terms, where n is odd, is equal to the negative of the sum of the n+1 terms.
the sum of 15 terms is -8
the sum of 17 terms is -9
the sum of 19 terms is -10
you can extrapolate from this to get the sum of 2013 terms as follows.
the sum of 2014 terms is 1007
therefore the sum of 2013 terms is -1007.
that should be your answer.
i confirmed using excel that this is the case so i'm reasonably confident this is the answer you are looking for unless i completely misinterpreted the problem.