Different approaches to 1+2+3+4+5+… = -1/12

All these are good, I think. Perhaps what needs adding, as a first reference point, is when we say, e.g.

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots = 2

this is not an addition sum. You can’t add infinitely many times, any more than you can live infinitely long or have infinitely many fingers.

It looks like an addition on the left-hand side. But what the left-hand side really means is that the limit of all the actual addition sums

1 + \frac{1}{2} + \frac{1}{4}

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}

1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}

(what are called the partial sums) is 2.

It’s not just a pedantic point. For example: by definition A + B = B + A. But if you rearrange the order of an infinite series (or, at least, of some infinite series) you can change the sum. For example:

S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} - \frac{1}{5} + \ldots = \ln 2

(it’s the standard series for ln (1+x) with x=1)

but the same series rearranged

= \left( 1 - \frac{1}{2} - \frac{1}{4} \right) + \left( \frac{1}{3} - \frac{1}{6} - \frac{1}{8} \right) + \left( \frac{1}{5} - \frac{1}{10} - \frac{1}{12} \right) + \ldots

= \left( \frac{1}{2} - \frac{1}{4} \right)  + \left( \frac{1}{6} - \frac{1}{8} \right) + \left( \frac{1}{10} - \frac{1}{12} \right) + \ldots

= \frac{1}{2} \left( \left( 1 - \frac{1}{2} \right)  + \left( \frac{1}{3} - \frac{1}{4} \right) + \left( \frac{1}{5} - \frac{1}{6} \right) + \ldots \right)

= ½ S

(click here for more info)

Thus when we write

1 + 2 + 3 + 4 + 5 + …

we have to read that not as an addition sum – it wouldn’t be an addition sum even if it were an “ordinary” convergent infinite series – but as “some calculation that makes sense of this infinite expression”.

The usual way we make sense of sums of infinite series – limit of partial sums – makes no sense with 1 + 2 + 3 + 4 + 5 + … But maybe there is another way to make sense of it? Yes, there is. That’s what these video clips are about.

Click here for a slideshow explaining what Ed Frenkel, in the last video, means by “regularised sums”.