calculate : fmax(n) = max(fmax(n-1)+value(n),value(n));

Why?
Consider p1 = 5, p2 = -2

fmax(1) = 5
fmax(2) = max( 5+(-2), -2 ) = 3

But if i = j = 1 fmax(2) = 5

Or I got wrong understanding of the task?

F[i] is max values of sequence with the final element is a[i].

Finally, answer will be max( F[1], ... F[n] ).
Sorry because of English.

It's wrong algorithm and more hard than right.