# Intuition for 1 + 2 + 3 + … = -1/12

You might have been led to believe the only way to make sense of the equation $1 + 2 + 3 + … = -\dfrac{1}{12}$ is with "zeta function regularization," involving analytic continuation or similar high-tech tools, and that there is no way to develop intuition for this.

You were probably was told this by *mathematicians,* to whom "intuition" means something very different than it does to me.

So here follows what I consider an intuitive explanation for this result. Prerequisites include an intuition for the arithmetic of positive integers (most readers will have this), and a willingness to stfu about rigor.

The "Counting" Function $C$
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For historical reasons, involving "counting" the number of individual objects (sheep, stones, sticks, whatever) in a collection, there exists a function $C$ that acts on ordered tuples of positive integers and outputs a single positive integer. For example,

$(1, 1) \xrightarrow[]{C} 2$ [^m7m8qjxhwwm]

$(1, 1, 1) \xrightarrow[]{C} 3$

Now, this function has several nice[^skujn4i2g7] properties (which have names like "commutativity" and "associativity"), which means that things like the following is also true:

$(2, 1) \xrightarrow[]{C} 3$.

I shall be dropping the superscript over the arrow for the rest of this.

*Fractions*, *Negative* numbers, *Zero*
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Now, playing with this function, we might ask questions like, 

*Is there a number *$x$* such that *$(2, x) → 1$*?* 

Or *Is there a number *$y$* such that *$(y, y) → 1$*?*

Now, obviously, the answer is "No, of course not. The output of the 'counting' function is greater than every number in the input, and there is no number smaller than 1." 

*However,* we're going to make them up in such a way that all the nice properties of the function $C$ hold; this is a silly exercise with no real-world meaning, but mathematicians like doing things like this, and they turn out to have surprising real-world applications, in fields like physics (something, something "unreasonable effectiveness"). These new "numbers" are called *negative numbers* and *fractions.*

$(2, -1) → 1,$ and $(\frac{1}{2}, \frac{1}{2}) → 1.$

Now, playing with these new "numbers," we find that to preserve the nice properties of $C,$ we need to introduce something called a "zero" or "0" such that $(1, -1) → 0$.

With me so far?

Infinitely Large Tuples and *Pattern Consistency*
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What happens if we apply our function $C$ to a tuple with infinitely many elements? The obvious answer is "That doesn't make any sense! You can't count infinitely many elements; you'll never get to the end. Consider Achilles and the tortoise …." Yes, of course, but like before, just go with it, okay?

The nice property we're going to make up for $C$ is that it follows patterns: in cases where the output isn't clear (such as with an infinite number of terms when we've only ever seen finite tuples), but there's a neat pattern that would be formed if it had some specific value, *that* is the value of the function. This is a little vague, but an example might make it clear:

$(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, …) → ?$

The pattern we're going to match is $(1)$, $(1, \frac{1}{2})$, $(1, \frac{1}{2}, \frac{1}{4})$, $(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8})$, and so on.

The output, following the pattern $1$, $1.5$, $1.75$, $1.875$, … is $2$.

The pattern is actually more general than that, and for any $r$, $|r| < 1$,

$(1, r, r^2, r^3, …) → \dfrac{1}{1 - r}.$

Here's the key idea:

*The outputs we've assigned using this pattern-matching technique can themselves be used as part of new patterns!*

For the tuple 

$(1, 2, 4, 8, 16, …) → ?,$

the analogous sequence $(1)$, $(1, 2)$, $(1, 2, 4)$, $(1, 2, 4, 8)$, etc. doesn't suggest any particular value. However, we can match it to the *metapattern* $\dfrac{1}{1 - r}$, where $r = 2.$

$(1, 2, 4, 8, 16, …) → -1.$

**Important caveat:** If there are multiple patterns that suggest *different* values for the output of a particular unclear tuple, this pattern matching approach won't work. That's fine, let's see how far it takes us!  Remember, if some patterns don't suggest any value,  that's not a problem as long as all the ones that *do* suggest the *same* value.

For some tuples, like $(1, -\frac{1}{2},  \frac{1}{3}, -\frac{1}{4}, …)$ the order of elements matters! Keep that in mind when matching patterns.  

Finish
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There isn't much more intuition involved to get from $(1, 2, 4, 8, 16, …) → -1$ to $(1, 2, 3, 4, 5, …)$. The hard part is finding the right patterns: many of the obvious things you might try don't suggest any value. After a lot of (other people's) work, here's an example of something that works.

$(e^{-1} \cos(-1)) →  0.1988,$

$(e^{-1} \cos(-1), 2 e^{-2} \cos(-2)) = 0.0861,$

 $(e^{-1} \cos(-1), 2e^{-2} \cos(-2), 3e^{-3} \cos(-3)) → -0.0617,$ …

and if you go far enough (use a computer!), the pattern suggests approximately $-0.08267.$

Now, we can use the results of these patterns to match a *metapattern:*

$(e^{-1} \cos(-1), 2e^{-2} \cos(-2), 3e^{-3} \cos(-3), …) → -0.08267,$$(e^{-1/2} \cos(-1/2), 2e^{-2/2} \cos(-2/2), 3e^{-3/2} \cos(-3/2), …) → -0.08329,$$(e^{-1/4} \cos(-1/4), 2e^{-2/4} \cos(-2/4), 3e^{-3/4} \cos(-3/4), …) → -0.08333,$ …

Notice that this pattern of infinite tuples *does* match what we want, $(1, 2, 3, …)$. The value suggested by this pattern is $-\dfrac{1}{12}$. □

It turns out no pattern that suggests a value suggests any other, though this is hard to prove.

Concluding Remarks
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 In *precisely* the same sense that we can write

$1 + \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{1}{8} + \dfrac{1}{16} + … = 2,$

despite that no real-world process of "addition" involving infinitely many terms may be performed in a finite number of steps, we can write

$1 + 2 + 3 + 4 + … = -\dfrac{1}{12}.$

It is only familiarity that keeps us from seeing this.

Further, "limit of partial sums" is only one of a whole class of "patterns" that may be used to assign values to infinite series. Its strength is that it's easy to describe in general, but it is not the only one, and where it fails to suggest a particular value in the limit, there are others that are no less reasonable.

[^m7m8qjxhwwm]: This is usually notated \(1 + 1 = 2\). Some people annoyingly include quotation marks around some of these symbols, but most people omit them for clarity. 

[^skujn4i2g7]: "Nice" is deliberately vague. We're gonna make up a bunch of elegant properties as we need them.