Maximum subarray is pretty well known problem with a simple algorithm to solve in O(n). Directly from the link above:

def max_subarray(A):
    max_ending_here = max_so_far = A[0]
    for x in A[1:]:
        max_ending_here = max(x, max_ending_here + x)
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

The problem mentions at least one positive integer, so this is equivalent to the above:

def max_subarray(A):
    max_ending_here = max_so_far = 0
    for x in A:
        max_ending_here = max(x, max_ending_here + x)
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

This is pretty simple to extend to a maximum window size k:

def max_subarray(A, k):
    max_ending_here = max_so_far = sliding_index = 0
    for i, x in enumerate(A):
        if sliding_index == k:
            max_ending_here -= A[i-k]
            sliding_index -= 1
        max_ending_here, sliding_index = max((x, 1), (max_ending_here+x, sliding_index+1))
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

Example:

In []: max_subarray([41, 90, -62, -16, 25, -61, -33, -32, -33], 5)
Out[]: 131
In []: max_subarray([-200, 91, 82, 43], 3)
Out[]: 216

Maximum subarray is pretty well known problem with a simple algorithm to solve in O(n). Directly from the link above:

def max_subarray(A):
    max_ending_here = max_so_far = A[0]
    for x in A[1:]:
        max_ending_here = max(x, max_ending_here + x)
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

The problem mentions at least one positive integer, so this is equivalent to the above:

def max_subarray(A):
    max_ending_here = max_so_far = 0
    for x in A:
        max_ending_here = max(x, max_ending_here + x)
        max_so_far = max(max_so_far, max_ending_here)
    return max_so_far

This is pretty simple to extend to a maximum window size k:

def max_subarray(A, k):
    max_ending_here = max_so_far = window = 0
    for i, x in enumerate(A):
        if window == k:
            max_ending_here, window = max((last_x, 1), (max_ending_here-A[i-k], window-1))

        max_ending_here, window = max((x, 1), (max_ending_here+x, window+1))
        last_x, max_so_far = x, max(max_so_far, max_ending_here)
    return max_so_far

Example:

In []: max_subarray([41, 90, -62, -16, 25, -61, -33, -32, -33], 5)
Out[]: 131
In []: max_subarray([-200, 91, 82, 43], 3)
Out[]: 216