I didn't think the Numberphile video on this was well done, in its "Math is a bunch of inscrutable magic you laypeople will never understand, and every attempt you (as represented by our camera man) make to voice your dumb intuitions is dumb, you big dumb dumbs!" way, but I even more was annoyed by the sneering "You're not allowed to do that! There are clear fixed rules and single, permanent, all-purpose definitions!" dismissals of the 1 + 2 + 3 + 4 + ... = -1/12 result in the backlash.

I'm going to copy and paste the explanation I originally wrote at Quora (https://www.quora.com/Whats-the-intuition-behind-the-equatio...), because I think it captures well everything I'd like to say about this at every level of the discussion:

The sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this:

First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and so, in some sense, X = 1/2.

Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and so, in some sense, Y = X/2 = 1/4.

Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and so, in some sense, Z = -Y/3 = -1/12.

In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't.

That's it. It's that simple. Everything else I'm going to say is just to comfort those who are uncomfortable with the game we've just played.

Note that I've said "in some sense" several times in the above argument. That's because, while we all know how to add and subtract a finite collection of numbers in the ordinary way, when it comes to adding and subtracting an infinite series of numbers, there are many different ways of interpreting what this should mean. Just knowing how to add finitely many numbers doesn't automatically tell us what it means to add a whole infinite series of them. And when it comes to summation of infinite series, it turns out there's not just one nice notion of "summation"; there are many different ones, which are nice for different purposes.

One such notion is "Keep adding things up, one by one, starting from the front, and see if the results get closer and closer to some particular value; if so, that value is the sum". On that account of what summation means, you clearly won't get any finite answer for 1 + 2 + 3 + 4 + ...; since the terms never get any smaller, the partial sums will never settle down to a finite value (and certainly not a negative one like -1/12!). They instead, in a natural sense, should be understood as summing to positive infinity.

And there's nothing wrong with this! You are not wrong to feel that 1 + 2 + 3 + 4 + ... is positive and infinite, and math does not deny this; there absolutely is an account of summation corresponding to this intuition.

It's just not the only account of summation worth thinking about.

We could instead consider other notions of "summation", including ones designed precisely so that arguments like the one we made at the beginning (which are very natural arguments to make!) counted as legitimate ways to reason about such "summation". And then, by definition, we will have that 1 + 2 + 3 + 4 + ... = -1/12, on such accounts of "summation".
(In doing so, we will lose certain familiar properties such as "A sum of positive terms is always positive". But this is how generalizations work; generalizations very often lose familiar properties. Even the textbook, limit-based account of infinite summation loses familiar properties like "The order of summation doesn't matter". Even finitary summation of integers loses the familiar property "If a sum is zero, so are all the summands" from basic counting. But there is a web of resemblances to more familiar kinds of summation which can justify, in certain moods, thinking of each of these generalizations as a form of summation itself.)

If you insist that "Keep adding things up and see if the results get closer and closer to some particular value" is the only account of summation you're interested in, you'll object to the argument we gave at the beginning, saying "You're not allowed to do that kind of shifting over and adding to itself reasoning all willy-nilly; look at what nonsense it produces!".

But it can be made sense of, and is even fruitful to make sense of, in certain contexts in mathematics, and there is no need to blind ourselves to this insight.

Again, that's it. It's that simple. Everything else I'm going to say is just to comfort those who are still uncomfortable. For those who want a more systematic, formal account of series summation of a sort which validates the above manipulation, read on:

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