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What is the sum of all the natural numbers ?

Posted 11 years ago
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 ?
POSTED BY: Colin KLINE
3 Replies
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)
See this or this Wikipedia articles for more details.
POSTED BY: Ilian Gachevski
Posted 11 years ago
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).
POSTED BY: Jari Kirma

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.

enter image description here

POSTED BY: Devendra Kapadia
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