# $\infty = -1$ paradox

I puzzled two high school Pre-calc math teachers today with a little proof (maybe not) I found a couple years ago that infinity is equal to -1:

1. Let x equal the geometric series: $1 + 2 + 4 + 8 + 16 \ldots$

$x = 1 + 2 + 4 + 8 + 16 \ldots$

2. Multiply each side by 2:

$2x = 2 + 4 + 8 + 16 + 32 \ldots$

3. Again from the equation in step 1, move the $1$ term to the left hand of the equation:

$x – 1 = 2 + 4 + 8 + 16 + 32 \ldots$

4. So the following appears to be true:

$2x = x – 1 \implies x = -1$

This is obviously illogical. The teachers told me the problem has to do with adding the two infinite geometric series, but they weren’t positive. I’m currently in Pre-calc, so I have extremely little knowledge on calculus, but a little help with this paradox would be appreciated.

When we talk about an “infinite sum”, we are really talking about a limit. In this case, we are talking about the limit of the “partial sums” of the series. The partial sums are:
\begin{align*} s_1 &= 1;\\ s_2 &= 1+2;\\ s_3 &= 1+2+4;\\ &\vdots \end{align*}
That is, $s_n$ is the sum of the first $n$ summands in the series. When we talk about the “value” of a series (an infinite sum), we are really talking about the limit of the $s_n$: that is, a specific real number $L$ that the $s_n$ are approaching as $n\to\infty$. Or we say that a series “equals $\infty$” if the values of $s_n$ grow without limit.

When you say $x = 1+2+4+\cdots$, what you are really saying is that the limit of $s_n$. In this case, the limit of the $s_n$ does not exist, because
$$\lim_{n\to\infty}s_n = \infty.$$
The values of $s_n$ get arbitrarily large as $n\to\infty$.

It is certanly true as well that the sum $2+4+8+\cdots$ is also $\infty$, since $2\times\infty = \infty$ (in the extended reals). And if you subtract one, then you still get $\infty$ because $\infty -1 = \infty$ (in the extended reals).

So you can write $2x = x-1$.

What you cannot do, however, is “subtract $x$ from both sides”; because that would be writing
$$2\times\infty – \infty = \infty -1 -\infty$$
and the problem is that even in the extended reals, $\infty-\infty$ is undetermined. It does not equal anything, and certainly not zero. In short, you cannot just cancel infinities.