Infinity Is Worth No More Than -1/12

On 16 January 1913, a confused manuscript reached the famous mathematician G. H. Hardy in Cambridge. Other researchers have received similar letters before, and rejected it due to the seemingly incoherent formulae mixed with trivial mathematical results. Professional mathematicians are used to receiving manuscripts by amateurs who believe to have solved famous problems, but this particularly odd scribble caught the eye:

1+2+3+4+5+6+\ldots+\infty=-\frac{1}{12}

Did this amateur mathematician really think that the sum of all natural numbers, a value that will exceed any given boundary at some point, will wind up being a negative fraction? M. J. M. Hill of the University College, London, simply responded that the author must have fallen victim to the pitfalls of divergent series and referred him to a standard textbook on the topic.

So, was this really just the work of a lunatic? Well, recently, the New York Times covered the topic, linking a video in which two physicists explain the importance of this result in modern string theory. While many physicists may not be too far from lunatics, these two make a pretty strong argument in this case: Continue reading Infinity Is Worth No More Than -1/12