This is in fact, not true. If you evaluate the sum of all the natural numbers (1 + 2 + 3 + 4...) it is infinitely large. To evaluate this sum to have a value of -1/12th is not really correct. =/
If you want to learn the mathematical reasoning behind why the answer is infinity, but why you could evaluate a similar looking sum to have a value of -1/12th, I suggest looking up the zeta function.If you have an interest in maths, I really recommend it, there are some surprising results and really beautiful mathematics to be found there. If you don't want to learn the maths, just take it as 1 + 2 + 3 + 4... = infinity

I'm not a mathematician, but according to wikipedia and wolframalpha, ?(?1) = -1/12

Mats1 is actually kind of correct. The thing is that the sum of the natural numbers is indeed infinite. In order to get -1/12 you have to use a different concept of numbers, called p-adic numbers.

For the curious, this question and answer give good a good introduction to the subject.

Anyway, this is further complicated by the fact that we aren't actually talking about 1 + 2 + 3 + ... = -1/12. What we're really talking about are limits and convergence, which isn't necessarily the same (or as strict) as equality. Because it's a limit we're computing, there are more ways to show 1 + 2 + 3 + ... = -1/12 than just the zeta function. So you might say 1 + 2 + 3 + ... is infinity just as much as you might say it's -1/12.

I don't know what 1+2+3+4+...=-1/12 has to do with p-adic numbers. Care to explain?

I'm not a mathematician, but according to wikipedia and wolframalpha, ?(?1) = -1/12

It might not be true under every system of mathematics, but it's certainly a correct answer under some of them. It's even used in physical calculations where the mathematical prediction matches up correctly with observations!

?(?1) = -1/12

This is the zeta function. Zeta of -1 DOES evaluate to - 1/12. The simple sum of writing 1 + 2 + 3 + 4 etc does NOT evaluate to - 1/12.

The zeta of -1 is the sum:

1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1

Isn't 2^-1 equivalent to 1/2, so 1/2^-1 would be 1/(1/2), which would be equivalent to 2? So 1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.

i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

Isn't it simply because they use different summation methods? I similar problem occurs when measuring an area with riemann integrals vs lebesgue integrals, some areas can be measured with one but not the other.

Isn't 2^-1 equivalent to 1/2, so 1/2^-1 would be 1/(1/2), which would be equivalent to 2? So 1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.

i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

It's because you can only do arithmetic on limits of series if they are absolutely convergent (i.e. the absolute value of each term, summed up, converges to a finite limit). Otherwise you can expect nonsense in general.

Isn't 2^-1 equivalent to 1/2, so 1/2^-1 would be 1/(1/2), which would be equivalent to 2? So 1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.
i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

The problem is that you can't sum infinitely many things. So when you write something like 1 + 1/2 + 1/4 + 1/8 + ... = 2, you are actually saying something like "the limit of the sequence (1, 1 + 1/2, 1 + 1/2 + 1/4, 1 + 1/2 + 1/4 + 1/8, ...) is 2". That is by far the most common way to interpret an "infinite sum", and according to that one 1 + 2 + 3 + 4 + ... = infinity.

There are other interpretations of "infinite sums". See the Wikipedia page on divergent series for details.

But writing the terms as k or 1/k^-1 shouldn't matter, should it? Same values in the same order, same series, right?

The zeta function is only equal to the summation if the argument (corresponding to the power in the sum) is greater than 1. (Well, it is also true if the argument is complex and the real part is > 1).

The series in the definition of the zeta function is convergent (in the usual sense) for complex numbers with real parts greater than 1. We may however give different definitions of this function with a greater domain and this analytic continuation is what we use to compute the zeta function of -1.