Question

What is the sum of all natural numbers to infinity?

Answer 1

There are a lot of different answers.

Explanation:

We can model the following.

Let `S(n)` denote the sum of all the natural number.

`S(n)=1+2+3+4+...`

As you can see the numbers get bigger and bigger, so

`lim_(n-> ∞)S(n)= ∞`

or

`sum_(n=1)^∞n=∞`

BUT, some mathematicians do not agree on this.

In fact, some think that according to the Riemann zeta function,

`sum_(n=1)^∞n=-1/12`

I do not know much about this, but here are some sources and videos for this claim:

https://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/

In fact, there is also a paper on this, but it looks pretty complicated to me. Anyways, here is the link for it.

http://math.arizona.edu/~cais/Papers/Expos/div.pdf

Answer 2

Ideas about `zeta(s)`

Explanation:

In higher level mathematics there is a specific function that is very closely associated with this sum, this is called: `color(blue)("Riemann Zeta Function")`:

Where `zeta(s) = sum_(n=1) ^oo n^(-s )`

So we see that `s = -1` yields the question you are asking...

`=> zeta(-1) = -1/12`

But there are also some very famous other series in mathematics:

`1/1^2 + 1/2^2 + 1/3^2 + 1/4^2+ ... = zeta(2) = pi^2 / 6`

But its very interesting to see how `1+2+3+4+ ...` supposedly converges to `-1/12`

But its well know that `1 + 1/2 + 1/3 + 1/4 + ...` actually diverges to `oo`

Few more interesting solutions of the riemann zeta function `zeta(s)`:

`zeta(-3) = 1/120`

`zeta(4) = pi^4 / 90`

`zeta(50)= (39604576419286371856998202 pi ^50) / 285258771457546764463363635252374414183254365234375`

"Values found on http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/03/ShowAll.html "