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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 ?
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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 ( and somehow manage to avoid telling that this type of a sum has a name: it's 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.

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POSTED BY: Devendra Kapadia
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