We call the step from x_n to x_(n+ 1) the "nth step". If | x _ (n+1) | =| x _ n |+1,then the nth step is called "upward", then x _ n > = 0；; Otherwise, | x _ (n+1) | =|x_n|-1,then this nth step is called "going down", then x_n < =0. You can draw an image, with the horizontal axis representing n and the vertical axis representing | x _ n | (note that the vertical axis represents absolute value instead of |) At this time, it can be seen from the figure that the so-called "a certain step is going down" means that the image goes to the next grid to the right at this point, not the previous grid. Although this graph represents the absolute value of x_n, we can see the positive and negative of x_n from the trend of this graph at this point. If you are upward, you are positive at this point, and vice versa.

Now we can see some sharp things in the picture. For example, if the successive terms in x_n are 2, 3, 4, -5, -4-3 (and then the absolute value of the latter term is 2), then these six terms are represented by red dots on this diagram. Sum these six items to get -3, and then throw them away from the original series to get a new series (that is, connect the right end of this part to the left end). The point in the diagram is upward, and if it is downward (for example, -5, -4, -3, 2, 3, 4), it is similar. This practice requires that there can only be one "vertex" in this "tip". In this example, the length of the tip is 6 (meaning it is 6 items) and the total is -3 (that is, the sum of these 6 items). For columns with 2n terms (for example, m, m+ 1, ..., m+n- 1, -(m+n), -(m+n- 1), ..., -(m+ 1).

After removing all the "tips", if the length of the sequence is reduced by 2n (from the previous method, the reduction amount must be even), then the sum of the removed 2n numbers is -n, leaving 2008-2n items. Note that it is impossible to remove all the points in the original sequence when eliminating the cusp (because the "right end point" will not be removed), so there are at least two items left (note 2008), so the remaining 2008-2n items form a straight line, so they will either go up all the time or go down all the time (if they go up for a while and down for a while, they will form a cusp, but I am considering the situation after removing all the "cusp". To sum up, there are two possibilities:

(1, go straight down) n= 1003, 2008-2n=2, and the absolute value of the remaining two items, the first item is 1 (the absolute value cannot be changed because it is connected in the process of removing the "tip"), and the latter item is 0. At this time, the first item must be-1 (in fact, even+1 is not important), because going to the right from it means "going down"; The sum of the removed 2n terms is -n, so the sum of the original sequences is -n- 1=- 1004.

(2) Go straight up, remove 2n items, leaving 2m=2008-2n items, and the remaining items form a straight line. At this time, the absolute value of the first item is still 1 (because the absolute value of the first item in the whole sequence is unchanged in the process of removing the "tip"), so the remaining items are 1, 2. The first 2m- 1 term must be positive, because this sequence is upward, and the last term can be positive or negative. At this time, the sum of the original sequence is-n+1+2+...+(2m-1)+(2m), that is,-n+2m 2+m or-n+2m 2-3m. Now 2m+2n=2008, so

-n+2m^2+m=2m^2+2m- 1004； -n+2m 2-3m = 2m 2-2m- 1004, where 2 < = m<= 1004.

Finding the minimum of the absolute values of the above two functions, it is not difficult to find that the first function when m=22 and the second function when m=23 can get 8, that is, in the case of (2), the sum of the whole sequence is the value closest to 0.

Summarize the two situations and get the answer. If there is anything that can be obviously simplified in this process, or there is no need to use such troublesome and imprecise intuition, please let us know downstairs. ?

PS？ Where to? 70 1 198:? The second line of your answer says X20082-x02 = 2 (x1+... X2007)+2008. Isn't this coming out soon? Why didn't you get the right answer? According to you, ((x _ 2008) 2-(x _1) 2+2 (x _ 2008)-2007)/2 is the value of x_ 1+ ... +x_2008. If we assume that x _ 2008 =+A? -? 1004, and then minimize the absolute value of this formula, right? However, in short, your idea is very concise.