Wow! What a great, detailed explanation. I understand the calculation of the sum of a geometric series, but I guess I'm still confused about the denominator.
if, d1=390 & d2=510, then r= d1/d2= .76
and your denominator is 1-r = .24
So are you saying difficulty rises by 24%? That's the part I don't get.
390 * (1+24%) = 484. This isn't 510.
The increase was (510-390)/390 = 30.7%
I'm sure I'm mixing something up.
if, d1=390 & d2=510, then r= d1/d2= .76
and your denominator is 1-r = .24
This is equivalent to saying that the difficulty rises by the same fraction each time it increases.
So are you saying difficulty rises by 24%? That's the part I don't get.
390 * (1+24%) = 484. This isn't 510.
The increase was (510-390)/390 = 30.7%
I'm sure I'm mixing something up.
Hmmm the common ratio itself, r represents the fractional decrease in mining profits with each diff change1
That is why you compare the two difficulties. So if the common ratio is .76, that means you will mine 24% less this fortnight than last fortnight yes. BUT don't make the leap with the 1/(1-r) thing. The 1/(1-r) term keeps tracks of the infinite sum, so I wouldn't bother even thinking of that as some fractional change.
Proof:
A = 1 + r + r2 + r3 + r4 + ...
= 1 + r ( 1 + r + r2 + r3 + r4 + ...)
= 1 + r A
A = 1 + r A thus
A = 1/ (1 - r)
You can see that there's a little infinity thing happening there so you do not want to worry about 1/(1-r) being some sort of fraction.
1 Note that your actual mining profits will not exactly feel this type of sharp dropoff...the actual way that works is more complicated but I don't think it is worth it to count the effects of such things. This model works alright despite its simplicity.
