There is a viral video, currently on the Net rounds, alleging that “the sum of of all the natural numbers = - 1/12”.
Anyone got a good “layman’s suitable” debunk for this ?
It really depends on what is meant by the “sum of all the natural numbers”. The series is divergent in the usual sense. Nevertheless, the statement is correct if understood in a regularization sense, i.e. looking at the analytic continuation of the Riemann zeta function to -1. Mathematica knows that
In[1]:= Zeta[-1]
Out[1]= -(1/12)
In[2]:= Sum[n^-s, {n, 1, Infinity}] /. s -> -1
Out[2]= -(1/12)
The videos (https://www.youtube.com/watch?v=w-I6XTVZXww and https://www.youtube.com/watch?v=E-d9mgo8FGk) somehow manage to avoid telling that this type of a sum has a name: it’s Ramanujan summation (http://en.wikipedia.org/wiki/Ramanujan_summation).
This was an interesting question and motivated me to write a blog post about the different methods for summing divergent series in Mathematica.
Unfortunately, writing the blog post took a little longer than expected due to my preoccupation with Mathematica 10 development, but it has now been published.
I hope that you will find the post useful.
