
The Ramanujan Summation: 1 and 2 and 3 and ⋯ + ∞ = -1/12 - agiri
https://medium.com/cantors-paradise/the-ramanujan-summation-1-2-3-1-12-a8cc23dea793
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fxj
The value is the analytic continuation of Riemann's Zeta function at the
argument -1.

[https://en.wikipedia.org/wiki/Riemann_zeta_function](https://en.wikipedia.org/wiki/Riemann_zeta_function)

There are several methods to define finite sums for divergent series. See the
wikipedia article for a comprehensive list:

[https://en.wikipedia.org/wiki/Divergent_series](https://en.wikipedia.org/wiki/Divergent_series)

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vivanchuk
Can't sum a divergent series

~~~
gus_massa
It's more complicated. The lie is using the same symbol for the usual
summation and the extended version of the summations.

We all agree about the value of a finite summation.

We all agree about the value of a sum when the summation is absolutely
convergent.

We all (almost) agree when the summation converges, but the convergence is not
absolute.

This is the standard definition that is used in a calculus course, but be
aware that when the convergence is not absolute there are some minor problems
here and there. It's much better if you have absolute convergence.

After that, the idea is that you can extend the definition of summation. The
more easy way is
[https://en.wikipedia.org/wiki/Cesàro_summation](https://en.wikipedia.org/wiki/Cesàro_summation)
but the other comment posted a link to a more generic article
[https://en.wikipedia.org/wiki/Divergent_series](https://en.wikipedia.org/wiki/Divergent_series)

Cesàro summation is fine, you don't get too many surprises using it. But when
you extend too much the definition of summation you get more weird results,
and you loose some properties of the standards summation, for example
a+b+c+d+e+... = a + (b+c+d+e+...).

In particular, to get a finite result for 1+2+3+4+... you have to use extend
the definition of summation a lot, perhaps too much, and there are many ways
to do the extension, and not all of them give the same result.

If you have half an hour to spare, I highly recommend to watch " _Ramanujan:
Making sense of 1+2+3+... = -1 /12 and Co._" by Mathologer
[https://www.youtube.com/watch?v=jcKRGpMiVTw](https://www.youtube.com/watch?v=jcKRGpMiVTw)

