The series is
-1, 2, -3, 4, -5, 6, ....
This seq till every even number can be considered in groups of two, each of which has a sum 1.
Thus, S(2) = 1, S(4) = 2, s(6) = 3 and so on.
But this seq's sum till every odd number is (seq no -1) /2 - the odd number.
Thus, S(3) = 1 -3 = -2, S(5) = 2 - 5 = -3 and so on.
From 1), Sum of all elements of the series is less than N.
This is true in all cases. Doesnt tell whether N is even or odd.
From 2), N has to be odd (see the above illustrations). For N even the sum is always + N/2, but is always -ve for odd N.
Who says elephants can't dance?