Simply put 1+1+1 = 3 , 2 + 4 + 6 = 12
1 + 1 + 1
0 + 2 + 4 + 6
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1 + 3 + 5 + 6 = 15
You can add any terms to any terms in a sum and get the same answer (added the 0 + to preserve indentation).
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Go for it... I'd welcome it. Just keep it simple
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Never said infinity = -1/12 ...
Maybe we can get another 50+ page thread (that's the last time I looked at it) about something that has no relevance to 99.999% (or is that 100%?) of people.
No. This is an abuse of notation.
For certain values, the Riemann Zeta function is equivalent to the geometric series. But when you analytically extend it, you're appending extra values to the Riemann Zeta. It is not the geometric series for these new values.
It's bad notation designed to confuse, used by lazy mathematicians and physicists who don't give a fuck. It pretty clearly flies in the face of the fact that the series diverges. But how can the series diverge and equal -1/12 at the same time? Because you aren't summing the series when you reach the -1/12 conclusion. It's just bullshit notation through and through.
The .999.... = 1 thing? Nobody disputes that. This shit? I took this to math professors at my institution (one of the strongest math programs), and they essentially all waved it off as crappy, nonsense notation.
OK, I'll try. It's not complete atm, but whatever.
So for Re(s)>1, the series Sum[n in N](1/n^s) converges. We want to analytically extend it to s=-1. To achieve this, we manipulate it: For Re(s)>1,
lim[N -> infinity] (Sum [n= 1, ..., N] (1/n^s) - 1 / ((s-1) N^(s-1)) - 1 / (2N^s) - s / (12 N^(s+1))) is the same (the latter terms converge to zero). We can extend this definition easily, for s=-1, we get: lim[N -> infinity] (Sum [n= 1, ..., N] (n) + N^2 / 2 - N/2 - 1/12) = -1/12.
Why these "arbitrary" terms? Because the whole thing can be extended easily (as a converging limit) to Re(s)>-2. For an account of the asymptotic behaviour of Sum [n= 1, ..., N] 1/n^s, as n -> infinity, you'd have to read through this for now (which is also why the terms have to be that way). I'll see if I can provide an easily digested "proof" of it.
@ Garnier: Yeah, it's really bad notation, but we know how it should be written down.
Also, I'm watching the video again and /sighhhhhhhh
"The sum is 1/2." No. No. The Cesaro sum is 1/2. By not explicitly mentioning that they're using a different summation method, they're essentially tricking people into believing that if you add 1's and -1's in the conventional way, you'll arrive at 1/2. Which is obviously false. Never mind that the essentially handwaved it after giving an intuitive argument.
And second, the addition of infinite series is which don't converge is itself ill defined in the sense that they're using it. For instance, if you go back to the 1 - 1 + 1 -1 ...... and you stick zeros in between every two numbers, you get a Cesaro sum of 2/3. So why don't Numberphile call the sum 2/3? I mean, it's the same thing in the simple way they're presenting it. Did I just prove that 1/2 = 2/3?
Numberphile isn't an expert on mathematics. He's a guy on youtube.
Somebody can make probably about a hundred grand a year with catchy viral videos. Whereas the sky is the limit if somebody is an actual expert at something.
So if he was an expert he wouldn't be on fucking youtube.
The guy needs to learn to brush his teeth before he does anymore of these videos.