Infinity and String Theory

There’s a Numberphile video that’s hurting people’s brains. It claims to prove (several ways, including in a companion video) that \[\sum_{n=1}^{\infty} n = -\frac{1}{12}\]

This is, of course, highly counterintuitive. The video itself is misleading in that the speakers refer to the sum of all natural numbers, when this is not in fact the sum of all natural numbers… and therein lies the problem, and (I think) why people are getting confused.

We tend to think of the infinite sum in this way: \[\lim_{n \to \infty} \sum_{k=1}^{n} f(k) = \sum_{n=1}^{\infty} f(n) \]

But this is not entirely valid. Summation is inclusive to both ends: \[\sum_{n=1}^{5} f(n) = f(1) + f(2) + f(3) + f(4) + f(5)\]

So: \[\sum_{n=1}^{\infty} f(n) = f(1) + f(2) + f(3) + … + f(\infty) \]

If \(f(\infty) = 0\) (that is, if the function converges to 0), then there’s no particular problem, and \[\lim_{n \to \infty} \sum_{k=1}^{n} f(k) = \sum_{n=1}^{\infty} f(n) \]

However, in the case of \(f(n) = n\), there is no convergence; \(f(n+1) > f(n)\) for all values of \(n\).

Put simply: Infinity is not a natural number. That should be a key takeaway of the Numberphile video, but they don’t even mention it (in fact, they imply exactly the opposite). \(\sum_{n=1}^{\infty} n\) is not the sum of the natural numbers, it’s the sum of the natural numbers and infinity. But infinity isn’t a number in the first place, at least not of the sort that people who are used to dealing with basic mathematics mean by “number”.

Another key point reflected in the Numberphile video is that \(\lim_{n \to \infty} \sum_{k=1}^{n} f(k)\) is not always equal to \(\sum_{n=1}^{\infty} f(n)\). This is a specific form of something that’s usually stressed repeatedly when the subject of limits is first introduced, namely that \(\lim_{n \to x} f(n)\) is not always equal to \(f(x)\).

Indeed, limits are very often used to find a meaningful surrogate for \(f(x)\) for those cases where \(f(x)\) is itself undefined. In the case of convergent sums, \(f(\infty)\) is generally undefined (by virtue of infinity not being a number), but if the function converges to 0, we can treat the limit of the sum as the sum itself.

That said, I don’t know much of anything about string theory, and I’m willing to accept the video’s claims that string theory has a use for this odd result. For standard mathematics, this is a case of “infinity breaks mathematics”.

Bonus Content (Tongue-in-Cheek)

So… let us accept the original claim. This allows us to calculate the numerical value of positive infinity! Since \(f(x) = x\) becomes increasing large as \(x\) approaches positive infinity, we also know: \[\lim_{n \to \infty} \sum_{k=1}^{n} k = \infty\]

In general: \[\left(\lim_{n \to x} \sum_{k=1}^{n} f(k)\right) + f(x) = \sum_{n=1}^{x} f(x)\]

So: \[\left(\lim_{n \to \infty} \sum_{k=1}^{n} k\right) + \infty = \sum_{n=1}^{\infty}n\]

We have two of these terms, and can replace thus: \[ \infty + \infty = -\frac{1}{12} \]

Hence: \[\infty = -\frac{1}{24}\]

And now we know.