Posted September 20, 2016


Sum1
1-1+1-1+1-1+1-1... = 0.5
Simply, at any given point, it's either 1 or 0, but since it's infinite you can't stop at any point, so you take the average, 0.5. Simple enough, right?
Sum2
1-2+3-4+5-6+7-8... = 0.25
How?
Take that series, add it to itself (but shift the lower numbers to the right for ease of understanding, so that you effectively get "1+0=1", "-2+1=-1", "3-2=1", etc) and you get the first sum (1-1+1-1+1...). So, 2*Sum2 = Sum1, and since Sum1 = 1/2, Sum2 = 1/4
Sum3 (What we're looking for)
1+2+3+4+5+6+7+8... = -1/12
Now, remove Sum2 from that.
1+2+3+4+5+6+7+8...
-[1-2+3-4+5-6+7-8...]
You get the series "1-1=0", "2-(-2)=2+2=4", "3-3=0", "4-(-4)=4+4=8", "5-5=0", "6-(-6)=6+6=12" and so on, so 4+8+12+16..., which obviously is 4*(1+2+3+4....), right?
Thus, Sum3-Sum2 (or Sum3-0.25) = 4*(Sum3), meaning -1/4=3*(Sum3), so Sum3=-1/12
And yes, these results are actually used in physics (for example, string theory, as mentioned in the video), though I'm far too much of a dummy to comprehend string theory so I can't explain how.
Post edited September 20, 2016 by Maighstir