Is the argument used by the Numberphile video to show that $1 + 2 + 3 + 4 + \dots = -1/12$ valid?

Question

At the end of the Wikipedia article on:

$$1+2+3+4 +\dots$$

an argument is present that the sum adds up to $-\frac{1}{12}$.

Here is the numberphile video.

Here's my attempt to fill in the details from the argument:

(1) Let $S_1 = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + \dots$

(2) Let $S_2 = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \dots$

(3) $2S_2 = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + \dots$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\, 1 - 2 + 3 - 4 + 5 - 6 + 7 + \dots$

(4) $\,\,\,\,\,\,\,\,\,\, = 1 -1 + 1 -1 + 1 - 1 + 1 - 1 + \dots$

(5) $S_1$ is the Grandi's series whose Cesaro's sum is $1/2$

(6) So, $S_2 = 1/4$

(7) Let $S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \dots$

(8) $S - S_2 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + \dots$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-1 + 2 - 3 + 4 - 5 + 6 -7 + 8 + \dots$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= 0 + 4 + 0 + 8 + 0 + 12 + 0 + 16 + \dots$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,= 4 + 8 + 12 + 16 + 20 + 24+ 28 + \dots$

(9) $4S = 4 + 8 + 12 + 16 + 20 + 24 + 28 + \dots$

(10) $S - S_2 = S - 1/4 = 4S$

(11) Solving for $S - 1/4 = 4S$ gives $S = -\frac{1}{12}$

Is this argument valid? Is there a flaw in the logic that leads to the right answer in the wrong way?

Answer 1

The argument is invalid for one simple reason:

The operator "$+$" is not defined for infinite sums.

---

Let me elaborate:

The symbol $+$ is a symbol that means something to us. It is a symbol that means "take whatever whole/rational/real/complex number is to the left of me and add it to whatever number is to the right of me."

In mathematical terms, $+$ is a mapping from $X\times X$ to $X$, i.e., if $X=\mathbb R$, then $+$ is a mapping that maps the pair $(x,y)$ into $x+y$. This would be more clear if we would write $$+(5,10) = 15,$$ but for historic reasons, we write $$5+10=15$$

Now, $+$ has some properties. For example, we know that $a+(b+c) = (a+b)+c$ for any three numbers $a,b,c$. This means that for any finite series of numbers $a_1,a_2,\dots a_n$, we know that no matter what the order in which we add the numbers, their sum is the same. That's why we are allowed to simply write

$$a_1+a_2+\dots + a_n$$

to denote the sum of the numbers.

However there is no trivial way in which we can define the sum of an infinte number of numbers!!!

The standard definition of a sum of real numbers is:

$$a_1 + a_2 + a_3 + \cdots = \lim_{n\to \infty} (a_1 + a_2 + \cdots + a_n),$$

and using this definition, the sum of all ones is $\infty.$

Answer 2

We need to be very careful with blind manipulations like the ones in the video - in particular, you can get situations like:

$$S = 1 + 2 + 3 + 4 + \dots$$

$$-S = S - 2S = (1 + 2 + 3 + 4 + \dots) - (2 + 4 + 6 + 8 + \dots)$$ $$= 1 + (2 - 2) + (3 - 4) + (4 - 6) + (5 - 8) + \dots = 1 + 0 - 1 - 2 - 3 - 4 +\dots = 1 - S$$

So $-S = 1- S$ and $0 = 1$.