let n - elements count, a(i) - your array f(i) - maximum sum of subarray that ends at position i (minimum length is 1). Then:
f(0) = a(i);
f(i) = max(f(i-1), 0) + a(i); //f(i-1) when we continue subarray, or 0 - when start at i position
max(0, f(1), f(2), ... , f(n-1)) - the answer
