# 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