### +1-1+1-1+1-1 . . . in cycles

One of the guys in the vid says there's a proof for Z (the Grandi series) = 1/2. I'll leave that to the more advanced, but to me that makes sense intuitively. In the absence of a graph for now, imagine a horizontal wavy line ascending with every +1 to a peak at 1 on the y axis and descending with -1 again to 0. Repeat, say with six 1's, so for three cycles. Now draw another horizontal straight line skimming along from peak to peak, and clearly you will have a figure falling into two equal halves above and below the wavy line. Each of these two halves will also equal the halves created by yet another horizontal straight line drawn from .5 on the y axis.

Now for the smart bit. Do an Oresme on the series, rearrange the +1's and the -1's into different, though still zero sum cycles: +1+1+1-1-1-1. This time the amplitude will rise to 3 and the horizontal through the mid points will be at 1.5, so the sum of the series will be 3/2.

This hocus pocus, if you want to call it that, also has an echo in nature. I've just been learning about Cepheid variable stars, and the close direct relationship between luminosity and the period over which luminosity varies from minimum to maximum and back. Luminosity, energy emitted per unit of time, could be represented by the area under the wavy line maybe. How the values 1/2 and 3/2 would relate to the natural parameters of luminosity and period I don't know, but any comments gratefully received.