1 − 1 + 1 − 1 + … = ???

[True Federalist (진정한 연방 주의자)]

The odd thing about Cesaro is that the order of what you sum matters.

Each of the below has the same number of + 1's and - 1's and a partial sum of 0 every fourth entry yet you get five different results

+ 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + ... = +½

- 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - ... = -½

+ 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ... = 1

+ 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + ... = 0

- 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 -  ... = -1

I'm not certain, but it looks like you can generate any rational number as the Cesaro sum with an appropriate ordering of the +1's and -1's that periodically has a partial sum of 0.  You could probably even generate irrational numbers with a defined aperiodic ordering of the +1's and -1's that had infinitely many partial sums of 0.  I wonder if anyone has come up with an algorithm to generate such an ordering for any desired real number.  Would be neat if they have.