What is the error in this fake proof which uses series to show that $1=0$?

Your argument hinges upon the assumption that $S$ is a number you can do arithmetic with. It isn't a number, you can't do arithmetic with it, and this is what you have shown (by contradiction).


To understand things like this, you have to pay careful attention to the underlying definitions. The definition of an infinite sum, like

$$1 + 1 + 1 + 1 + \cdots$$

is the limit

$$\lim_{n \rightarrow \infty} \underbrace{1 + 1 + \cdots + 1}_{n}$$

i.e. the sum of $n$ ones, as $n$ is allowed to approach infinity. However, this limit does not exist in the real number system, because the right-hand term grows indefinitely large.

Yet, by substitution, this limit is the value you have decided to represent by the symbol $S$. Your problem, then, is that such a value does not exist. The sum of the infinite series doesn't exist. Hence $S$ has no referent, and the associated computations are meaningless.

That said, an alternative, and perhaps stronger, perspective would be to say that if an object like $S$ existed, and it permitted the manipulations you did, it would break things, because its existence would thus embody contradictions.


Of course you may be wondering, then, "but what about $\infty$? Isn't

$$\lim_{n \rightarrow \infty} \underbrace{1 + 1 + \cdots + 1}_n = \infty$$

?"

The answer is: no, not in the real number system. In the real number system, the limit does not exist. The above equation is often shown, but its meaning is not really made clear. What it "really" means is an equation in the extended real number system, where an additional element called $\infty$ has been added, and that results in the prior limit as being valid. In that case, then yes, $S = \infty$. Yet, given the last paragraph of what I just said above, something has to break for this not to be contradictory. What breaks is that $\infty$, as an extended real number, but not a real number. And once allow $S$ to take extended-real values, the very rules of algebra change, as you are working in a different number system - it is like going into the complex numbers by adding $i$. Namely, in the extended real numbers you are not allowed to start with

$$S = 1 + S$$

then "subtract from both sides"

$$S - S = (1 + S) - S$$

and then "cancel". The subtraction is okay, but not the cancellation. You now cannot infer that the left-hand side is zero. In fact, $\infty - \infty$ is, itself, undefined, in this extended real number system.

If you go this route, what you learned in grade school quits working.


You are treating infinity as if it were a number. However, it is not, and therefore you cannot perform ''usual'' operations such as $+$ and $\times$ on it.