The max of all possible sequential sums - excel

Question

Is there a way to write this using formulas.
=Max(a, a+b, a+b+c, a+b+c+d, a+b+c+d+e+...)
I don't want to use VBA for this task and I am not sure how to approach this problem.
Excel sheet expanded with formula
Excel sheet contracted without formula
Objectives:
1) For each of the peoples, (Bill, Ben, Katy), I would like to compare the maximum of the sum of only X through time.
For example, in Column J, I would like to know the current Max of the sum of X. Current is the date 1/17/19 because it is the most recent entry.
2) For each of the peoples, (Bill,Ben, Katy), I would like to compare the max of the previous entry to the max of my most recent entry.
For example, in Column K, I would like to compare the Max of the sum of X at 1/5/16 to the max of the sum of X at 1/17/16.
3) I would like Column J and K to recalculate as I bring in new data entries into Column I. As of now, using a solution mentioned below, in Column J, I think I would be using something like this formula:
=MAX(MMULT(0+(ROW(B9:I9)>=TRANSPOSE(ROW(B9:I9))),B9:I9)
This solution seems to work if I only have X's going down vertically though.
Also, as new data gets brought in, Column J and K would be pushed to the right, becoming Column K and L.
4) The highlighted region in Column I and J are my output sections that are dependent on the date of Cell B1.
For example, if I were to change the date to 1/1/2016 in Cell B1, Cell I7 would equal -4 Cell J7 would equal True.
If I were to change the date to 1/17/16 in Cell B1, Cell I7 would equal 5 and Cell J7 would equal True.
I've been playing around with this a bit, trying to use SUMIFS to pick up sum of X based on a date criteria.
I thank everyone in advance for all your help, and I apologize if my wording to this problem is unclear. I am a undergraduate student, and have no background in computer/programming/anything of that sort at all. Thank you so much!

Answer 1

Assuming a, b, c etc are in a column, then a standard approach will be to use an additional column with a cumulative sum of the values in the input column. Then you can just take the MAX of the column with the cumulative sum.
E.g. use of cumulative sum in column B with the input values in column A:
With MAX formula in column C:

Answer 2

Assuming data in A1:A5, you can use this array formula**:
=MAX(MMULT(0+(ROW(A1:A5)>=TRANSPOSE(ROW(A1:A5))),A1:A5))
which, due to its being non-volatile, is preferable to:
=MAX(SUBTOTAL(9,OFFSET(A1:A5,,,ROW(A1:A5)-MIN(ROW(A1:A5))+1)))
Regards
**Array formulas are not entered in the same way as 'standard' formulas. Instead of pressing just ENTER, you first hold down CTRL and SHIFT, and only then press ENTER. If you've done it correctly, you'll notice Excel puts curly brackets {} around the formula (though do not attempt to manually insert these yourself).