1 + 2 + 3 + 4 + …

There’s a video that has gone viral and concerns itself with the equation

1 + 2 + 3 + 4 + \ldots = - \dfrac{1}{12}.

A couple of students from my Calculus 2 class last semester sent me the following e-mail, which makes me smile for several reasons:

We are rather skeptical, but do not really understand explanations of the “proofs” others have provided. Since we consider you the expert on convergence and divergence we were hoping you knew what the answer actually is. Is this a result that’s accepted by the mathematical community? Or are we just being lied to by the internet?

I think Calculus 2 (which includes a fairly detailed coverage of Taylor series) is a great moment to be asking this question; here was my response:

The very short answer to your question is, no, it is not really true that the sum of the positive integers is equal to −1/12.  But there is an interesting piece of mathematics that lies behind that equation, and it’s not quite the total nonsense it appears on first glance.  I recommend the following blog post by a professor at Utah:

Here is my supporting commentary to go along with it.  The mathematical context called is the field called “complex analysis” — it’s essentially what you get if you do calculus, but you use the complex numbers as your basic set of numbers instead of the real numbers.  Many things work out exactly the same; for example, the definition of the derivative is exactly the same, except that the little h going to 0 is now a complex number instead of a real number.  And one can still build up all the machinery of Taylor series in this context.  Now, we’ve seen that sometimes Taylor series converge only in some cases.  For example, we know
1+x+x^2+x^3+\ldots = \dfrac{1}{1 - x}
but only for certain values of x (namely |x| < 1), even though the function on the right is defined in other situations (for any x other than 1).  So, the equation you’re asking about is similar to
1+2+4+8+16+\ldots = -1

with the “proof” that you put x=2 into that formula for geometric series above.  This is not right (we’re outside the interval of convergence, and the series on the left diverges), but it’s possible to understand what is meant by the equation.  The 1+2+3+4+… example is similar, but with a more sophisticated series.

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