

1+2+3+4+ . . . = -1/12 - sz4kerto
https://www.youtube.com/watch?v=w-I6XTVZXww

======
ColinWright
This has been discussed a lot, but there is something real and genuine going
on here that is, unfortunately, masked by the obvious nonsense.

Briefly let me introduce the main ideas.

When -1<x<1, sum_{n=0}^{n=oo} x^n = 1/(1-x). We might be tempted, then, to say
that

    
    
        1 + x + x^2 + x^3 + x^4 + ... = 1/(1-x)
    

But that's obvious nonsense when, say, x=2. The LHS clearly makes no sense,
even though the RHS exists and has the very sensible value of -1. It therefore
cannot be said that:

    
    
        1 + 2 + 4 + 8 + 16 + ... = -1.
    

However, underlying all this is the concept of _Analytic Continuation_ which
says that under certain circumstances, this is a reasonable thing to be doing.

Sort of.

Here it is being done properly:

[https://terrytao.wordpress.com/2010/04/10/the-euler-
maclauri...](https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-
formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-
continuation/)

Here is a Hacker News discussion of that:

[https://news.ycombinator.com/item?id=7078744](https://news.ycombinator.com/item?id=7078744)

If that has whetted your appetite for more you might like to follow some of
the discussions and links given in previous submissions:

[https://news.ycombinator.com/item?id=7078489](https://news.ycombinator.com/item?id=7078489)
: 53 comments

[https://news.ycombinator.com/item?id=7073976](https://news.ycombinator.com/item?id=7073976)
: 29 comments

[https://news.ycombinator.com/item?id=7057049](https://news.ycombinator.com/item?id=7057049)
: 19 comments

[https://news.ycombinator.com/item?id=7038809](https://news.ycombinator.com/item?id=7038809)
: 6 comments

[https://news.ycombinator.com/item?id=7081885](https://news.ycombinator.com/item?id=7081885)
: 4 comments

[https://news.ycombinator.com/item?id=8843219](https://news.ycombinator.com/item?id=8843219)
: 4 comments

Other submissions:

[https://news.ycombinator.com/item?id=7033444](https://news.ycombinator.com/item?id=7033444)

[https://news.ycombinator.com/item?id=7074820](https://news.ycombinator.com/item?id=7074820)

[https://news.ycombinator.com/item?id=7079921](https://news.ycombinator.com/item?id=7079921)

[https://news.ycombinator.com/item?id=7096326](https://news.ycombinator.com/item?id=7096326)

[https://news.ycombinator.com/item?id=7176024](https://news.ycombinator.com/item?id=7176024)

[https://news.ycombinator.com/item?id=7602317](https://news.ycombinator.com/item?id=7602317)

[https://news.ycombinator.com/item?id=7638147](https://news.ycombinator.com/item?id=7638147)

