I can imagine asking them to think about a finite number of terms in this series:

M(k) = 1 + 2 + 3 + 4 + … + k

(k is a positive integer)
I’d be very comfortable showing them that a compact way of writing this sum is:

M(k) = k(k+1)/2.

What would you recommend to say at this point, then, to help them understand the sense in which M(k) is equal to -1/12 if you let k grow to infinity?

(Again, I’m thinking about explaining this to a middle schooler and their teacher. As you point out, the tricks you use here are hard to justify because other seemingly similar tricks yield different answers. But the point of this problem is much more important than showing how to ‘prove’ that 2 = 4.)

As an aside, if I talked with them about the sum:

N(k) = 1 – 1 + 1 – 1 + … + (-1)^(k-1)

I could show them that

N(k) = [1 + (-1)^(k-1)]/2

which is 1 for odd values of k and 0 for even values. Telling them, then, that we will define the sum as equal to 1/2 for infinite k might seem reasonable in the sense that 1/2 is the average of 0 and 1.

I’m not disputing the importance of saying that

M(k) = k(k+1)/2 -> -1/12,

I’m just curious about the suggestion that this is something to share with middle schoolers and teachers. Is there a way to help them understand this in the context of middle-school level arithmetic?

Divergent series can be manipulated to give lots of funny answers.

I can imagine asking them to think about a finite number of terms in this series:

M(k) = 1 + 2 + 3 + 4 + … + k

(k is a positive integer)
I’d be very comfortable showing them that a compact way of writing this sum is:

M(k) = k(k+1)/2.

What would you recommend to say at this point, then, to help them understand the sense in which M(k) is equal to -1/12 if you let k grow to infinity?

(Again, I’m thinking about explaining this to a middle schooler and their teacher. As you point out, the tricks you use here are hard to justify because other seemingly similar tricks yield different answers. But the point of this problem is much more important than showing how to ‘prove’ that 2 = 4.)

As an aside, if I talked with them about the sum:

N(k) = 1 – 1 + 1 – 1 + … + (-1)^(k-1)

I could show them that

N(k) = [1 + (-1)^(k-1)]/2

which is 1 for odd values of k and 0 for even values. Telling them, then, that we will define the sum as equal to 1/2 for infinite k might seem reasonable in the sense that 1/2 is the average of 0 and 1.

I’m not disputing the importance of saying that

M(k) = k(k+1)/2 -> -1/12,

I’m just curious about the suggestion that this is something to share with middle schoolers and teachers. Is there a way to help them understand this in the context of middle-school level arithmetic?