TITLE: Explain non-convergent sums to a bright high schooler.
QUESTION [26 upvotes]: I know the higher-math answer to this question, but I'm asking how on Earth to explain it to a bright high school student.  Here's the question, paraphrased:
“I can see that $\left(1+x+x^{2}+x^{3}+x^{4}+....\right)\left(1-x\right)=1$ because the terms with non-zero powers of $x$ all cancel out.  But if I plug in $x\rightarrow2$, the left factor is $(1+2+4+8+….)$ and the right factor is $-1$.  So how can positive times negative equal positive?  How can infinite times finite equal finite?  This makes no sense!  Math doesn't work!”
When I attempt to explain non-convergence or limited domains of convergence, the sharper students react with something like, “What the hell, math just decides to not work sometimes???  So how do we know when we're supposed to disregard logical conclusions???  I no longer trust adults!!!”
Any advice for a stumped tutor?  “You’ll understand when you’re older" and "How can we have a unified country if you don't believe the things we tell you?" are not acceptable responses.

REPLY [3 votes]: I think the existing answers are pretty unconvincing or hard to read, especially in light of the whole -1/12 business. 
Here's what I think is a better approach:
Point out to him that the "reason" we have
$$1 + x + x^2 + \ldots = \frac{1}{1 - x}$$
is NOT that the terms "cancel". 
Rather, it's that if the left side had a finite sum $S$
$$S = 1 + x + x^2 + \ldots$$
then we can prove that $S$ must satisfy the following:
\begin{align*}
S x &= x + x^2 + x^3 + \ldots \\
1 + S x &= 1 + x + x^2 + \ldots  \\
1 + S x &= S && \text{ (true by our assumption)}  \\
(S - S x) &= 1  \\
S (1 - x) &= 1  \\
S &= \frac{1}{1-x}
\end{align*}
However—and this is key—we have merely proved a CONSTRAINT on such a sum $S$.
We did NOT prove that $S$ exists at all! We simply assumed it and derived a property that $S$ satisfies.
To prove that the the two sides are equal, then, we must first define what it means for a finite series to be "equal" to an infinite series, since, until now, we've only had a meaning for equality for finite expressions.  
Of course, a very reasonable definition for this (which I assume they have been taught) is that the partial sums of the series converge to a limit $S$ (using the epsilon/delta definition of a limit).
And of course, once we use that definition, then all you need to do is show that the partial sums do (or don't) converge to $S$, and only after doing that can we declare $S$ to be equal to $1 + x + x^2 + \ldots$.
[Optional] However, if you change your definition of equality of a finite and an infinite series, you can obtain a different result. For example, if you simply define the sum $S$ of an infinite series to be "whatever number $S$ that satisfies the same algebraic expressions as the original series", then the student's statement will be correct. You should point out that this is not the definition of equality they have been taught, however, as there are many pitfalls to this approach, and it is useless (if not outright harmful) for most practical purposes, including the the fact that—as he rightly pointed out—such a definition would violate many intuitions.
Nonetheless, I think if you explain the above clearly, he should be able that clearly see that the math he has already learned works either way, and that there is no contradiction whatsoever.
The only contradiction comes up if he decides to define equality differently than you have.
Which is fine, because nobody guaranteed different definitions would lead to the same conclusions!
(P.S.: This is also a good way to teach why definitions matter.)