A different Harmonic series.

Solution 1:

This is A232111(n)/A232112(n):

1/1, 1/2, 1/6, 1/12, 7/60, 1/20, 11/420, 13/840, 11/2520, 11/2520, 23/27720, 23/27720, 607/360360, 251/360360, 251/360360, 25/144144, 97/12252240, ...

Solution 2:

If it helps, these are the answers for the first few cases. "1" stands for adding and "-1" for subtracting. For instance, $[1,1,-1]$ would mean the number ${1\over 1}+{1\over 2}-{1\over 3}$. The best choices are $$[-1, 1], [-1, 1, 1], [-1, 1, 1, 1], [1, -1, -1, -1, 1], [-1, 1, 1, 1, -1, 1], [1, -1, -1, -1, -1, 1, 1], [-1, 1, 1, 1, -1, -1, 1, 1], [-1, 1, 1, -1, 1, -1, 1, 1, 1], [1, -1, -1, -1, 1, 1, -1, -1, -1, 1], [-1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1], [1, -1, -1, -1, 1, -1, 1, 1, -1, -1, -1, 1], [-1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1], [-1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, 1, 1, 1]$$ I used a program to find them, and I'm pretty sure they are accurate. The sums are $$0.5, 0.16666666666666669, 0.08333333333333331, 0.1166666666666667, 0.04999999999999996, 0.02619047619047618, 0.015476190476190477, 0.00436507936507935, 0.004365079365079377, 0.000829725829725847, 0.0008297258297258053, 0.0016844266844266292, 0.0006965256965257016$$

I have added a pictorial representation below. That may help to find a pattern.

-+
-++
-+++
+---+
-+++-+
+----++
-+++--++
-++-+-+++
+---++---+
-+++---++++
+---+-++---+
-+-++-++++-++