Some time ago I blogged here about the absurdity of the infinite sum 1+2+3+4… = -1/12. I did a partial analysis here but I never really completed my thoughts. I’m lazy.
Anyway I noticed today a great explanation by Mathologer on youtube. I wont explain here his logic but it closely matches what I was thinking at the time. Go see his video. Basically the idea that the infinite series 1-1+1-1+1-1… is a number at all is simply wrong. So using it to determine the sum 1+2+3+4… is wrong. As I said at the time, maths getting derailed.
What is curious is how we can use numbers and operations on numbers to construct something that is not a number. There is no problem with this. Most agree that 1/0 is not a number. Some would argue the ‘number’ is infinity. OK.. but does 1/0 + 1/0 = 2 x infinity? No. Regular numeric operators (+, -, *, /) don’t apply. So if you sum s = 1-1+1-1+1… doesn’t equal a number, you can’t expect adding and subtracting other infinite series from it makes sense.