The Sum of All the Natural Numbers

[Dadface]

Look at how he adds S2 to S2. He writes one series of numbers (from 1 to 7 in the example given)and writes an identical second series of numbers which he places below the first series.He then pushes one of the two series along by one and concludes that the sum is given by 1-1+1-1 etc .The sum, however, is not given by that. In the example given where the last digit equals 7 the sum is equal to 8. He ignored the seven at the end of the pushed along series. This number 7 was not overlapping with any other number.
When the non overlapping number at the end of the series is taken into account one can see that 2s2 approaches infinity as the number of numbers in each series approaches infinity (plus or minus infinity depending on the number of digits). 2s2 has numerical values which increase with the length of the series as follows:
+2,-2,+4,-4,+6,-6, etc

 

[atyy]

atyy, Cesaro summation is a rigorous method of assigning values to series. The series 1+2+3+... is not Cesaro summable though, so it doesn't help with the original problem. It is used in some areas to great effect; for example partial sums of Fourier series can be quite bad on continuous functions even (diverging everywhere), but the Cesaro sum ends up converging for functions that are way worse than continuous. It ends up that when you want to prove results about Fourier transforms it's a lot better to consider the Cesaro sum of the Fourier series rather than the regular sum.

Notice how the wikipedia article then goes on in a later section to describe a whole host of issues in which inserting zeroes into sums can change the Cesaro sum, so when doing Cesaro summation you have to be extremely careful that you are being actually rigorous, and not just taking the word rigorous and slapping it onto a bad argument.

Is it right to say that 1+2+3+... is also not Abel summable?

So one really has to use zeta regularization, or the smooth cut-off mentioned in the link that D H gave in #56?

 

[pwsnafu]

Is it right to say that 1+2+3+... is also not Abel summable?

Correct. It is mentioned in Wikipedia page as well.

 

[1MileCrash]

'After taking a course in mathematical physics, I wanted to know the real difference between Mathematicians and Physicists. A professor friend told me "A Physicist is someone who averages the first 3 terms of a divergent series"'

I thought you'd all enjoy that one.

 

[AlfieD]

'After taking a course in mathematical physics, I wanted to know the real difference between Mathematicians and Physicists. A professor friend told me "A Physicist is someone who averages the first 3 terms of a divergent series"'

Hmmm, interesting!

 

[pjantoniopj]

Here's where i saw the first mistake with Numberphile;

Given:
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

Numberphile folks concluded that they could not just do the regular math (addition, subtraction) to obtain S1 = 0.
So they introduced their next step; add S1 to S1(shifted to the right)

S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...
S1 = 1 - 1 + 1 - 1 + 1 - 1 + ...

And then, magically, they concluded that Yes, now we can to regular math here. They proceeded to add and subtract in a vertical direction.
Wait, what? You couldn't add and subtract horizontally but you can add and subtract vertically?

If you can add 1+0, then -1+1, then 1-1, then -1+1, etc., FOREVER, then why couldn't you do that in the first place with just S1?

 

[cryptist]

Algebraic manipulations on the video may be wrong, however, it is TRUE that finite and physically useful results can be obtained from divergent sums. In fact, mathematicians themselves are using it. There is a technique called analytic continuation that provides an extention of the domains of analytic functions. Note that, these odd results also consistent within the complex analysis. These divergent sums do not have contradictory finite results, they are actually valid!

Secondly, Casimir effect is a physical phenomenon that experimentally verified and well understood. In these effect, infinite summation of frequencies gives Zeta[-3]=1/120 where Zeta[-3] is a divergent sum and this result is verified experimentally. So in nature infinite summation of some "things" can REALLY give a finite result, which is consistent with mathematics.