A bit over a year ago, I wrote a blog post about the mathematics of infinite series, and how weird such series can be, considering in particular the behavior of "conditionally convergent series". A recent post at Built on Facts covered similar oddities and gave a nice and different perspective on them. In the comments of that post, though, an even more bizarre result from the theory of infinite series was introduced, namely the argument that

$latex displaystyle 1+2+3+4+ldots = -1/12$.

This result, if true, is enough to shake one's faith in mathematics, and is completely non-intuitive for no less than three very big reasons:

- The sum of an infinite series of increasing
*positive*integers should not converge to a*negative*value, - The sum of an infinite series of increasing positive
*integers*should not converge to a*fractional*value, - The sum of an infinite series of
*increasing*positive integers should not converge at all to a*finite*value!

So is the equation above correct? Not exactly; it is based on a valid bit of mathematics centered on the Riemann zeta function, but that mathematics is being somewhat misinterpreted to get the paradoxical equation. An explanation of what went wrong is interesting in itself, however, and allows me to describe a rather difficult concept in the theory of complex analysis known as *analytic continuation*.