Physics contradicts maths - how is this possible?

As someone who has had some training in analysis (the branch of mathematics that deals with limits), I can offer an explanation of your paradox.Neither of the series has a limit that is defined, and so neither x nor 2x are defined. Therefore, the subtraction doesn't make sense, since neither x nor 2x represent real numbers.Edited by Chiroptera, : typo

---

It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

I should add that if we put a different metric on the rationals, the 2-adic metric, then the completion is what we call the 2-adic numbers. In that casex = 1 + 2 + 4 + 8 + ...does have a limit, and indeed x = -1 in the 2-adic numbers.This is easy to see if I may offer this proof (it is left to the reader to supply the more rigorous steps):x = -1 if 1 + x = 0 (this is what -1 means, after all).1 + x = 1 + 1 + 2 + 4 + 8 + ...
= 2 + 2 + 4 + 8 + ...
= 4 + 4 + 8 + ...
= 8 + 8 + ...and so forth. You notice that at each step the addition "knocks out" the next higher power of two, and "eventually" (defined in a suitably rigorous limit kind of way) nothing is left, so, indeed, 1 + x = 0 and so x = -1.The p-adic numbers have all sorts of weird topological properties that make them fun, which is why I know a little about them.Edited by Chiroptera, : another typo

---

It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

That I did not know.

---

It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

No, I would first show that the sequence of partial sums is monotonic and bounded (this would be pretty easy) -- convergence then follows trivially, and then Modulus' method suffices to prove the value is indeed 1.

---

It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

Actually, there are a lot of series and sequences where the actual limit isn't obvious from the form of terms. In that case, it's a pretty standard technique to first show that it converges, and then use algebra tricks like Modulus' to find the actual limit. These sorts of things especially common on exams; in real life (that is, real mathematics papers), the actual value of the limit is often of little interest -- what is of interest usually just convergence, and showing convergence is often a lot easier than figuring out the actual value of the limit.Anyways, since you just mentioned that it's a geometric series, the easiest thing to do is just use the geometric series test to show that it converges, and then the geometric series formula to evaluate the limit. No need to even discuss limits.

---

It has become fashionable on the left and in Western Europe to compare the Bush administration to the Nazis. The comparison is not without some superficial merit. In both cases the government is run by a small gang of snickering, stupid thugs whose vision of paradise is full of explosions and beautifully designed prisons. -- Matt Taibbi

What? A geometric series? This is basic Calculus II. Or showing that a sequence is monotonic and bounded? This is undergraduate elementary analysis.- As a matter of fact, that turns out to be my job.Do you know anything about mathematics?

---

"The guilty one is not he who commits the sin, but the one who causes the darkness."
Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo

*shrug*I dunno. Maybe they were trying to force you to learn the basic principles. Me, I'm not going to try to second guess other peoples' teaching methods.

---

"The guilty one is not he who commits the sin, but the one who causes the darkness."
Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo

Bwahahaha! Are there still constructionalists? I thought they all died off with the people who don't accept the Axiom of Choice.Oh, wait a minute. I know someone who doesn't accept the Axiom of Choice. Never mind.

---

"The guilty one is not he who commits the sin, but the one who causes the darkness."
Clearly, he had his own strange way of judging things. I suspect that he acquired it from the Gospels. -- Victor Hugo