There is no rule that justifies the OP’s procedure. Cesaro summation does work on it, however, but that gives an average, not a real value for the series. The real surprise is that rearrangement of a (conditionally) convergent series can make it converge to any sum you wish or to diverge. (It follows, by the way, that a suitable rearrangement of a divergent series might force it converge.) The series 1 + 1 - 1 + 1 + 1 - 1 + 1 + 1 - 1 + … is a rearrangement of the series of the OP (it has infinitely many 1’s and -1’s doesn’t it) but goes to infinity. The convergent–but not absolutely convergent–series 1 - 1/2 + 1/3 - 1/4 + 1/5 - … does converge in the usual sense to ln(2), but a suitable rearrangement can be made to converge to pi or e or 7 or go off to infinity. The point is that that the sequence of absolute values 1 + 1/2 + 1/3 + 1/4 +… diverges.
How do you obtain 1 - 2 + 3 - 4 + … = 1/4 via “shifting”?
S = 1-2+3-4+…
So S+S =
1-2+3-4+5…
- 1-2+3-4
2S = 1-1+1-1+1
1-1+1-1+1 can be shown to be equal to 1/2 (see my OP)
So. . .
2S = 1/2
S = 1/4
n/m; I assume this is done by reducing to the series in the OP and proceeding analogously. However, as has been pointed out, without further analysis and qualification “shifting” is not rigorous and can lead to apparently contradictory results.