Physics contradicts maths - how is this possible?

In terms of quantum physics, which you would have to take into account if you are dealing with the minute scales indicated by the problem, the concept of "reaching" a "point" can only be defined through statistical significance. Unlike in the deterministic mathematical model described in the OP, the statement that a particle has reached a "point" only says that there is a probability that a measurement will find the particle at that "point".

It's not an acceptable mathematical proof, because you are assuming certain facts about the convergence of the series and convergent series in general.That is why kongstad went to such lengths.

x = 1 + 2 + 4 + 8 + 16 + ....2x = 2 + 4 + 8 + 16 + ....2x - x = -1 (subtracting terms and "going all the way").x = -1That is what can go wrong and why your "proof" is mathematically invalid.

That is correct. So you first have to prove convergence which is the point I was making.

You can't assume you have a real number until you show your sequence converges. You can't talk of x or x - x until you have shown it is a real number.My example showed what can go wrong if the sequence diverges. Also, "2 x infinity = infinity" is not a valid mathematical equation, so no proof there.You need to open a book on mathematical analysis. You are murdering the subject.You need to first understand the field axioms and what constitutes a logical mathematical argument.

Yes it does. It could mean the sequence diverges. Proving it doesn't is mathematics.'Not being sure how one could argue otherwise' is not mathematics.

Physical contact does not necessarily imply travelling at the same speed. Physical contact means significant intermolecular interaction at molecular scales. The molecules of the fly are decelerated to zero by intermolecular forces as they approach the molecules of the train. The fly molecules go through zero velocity while the train is still approaching them, because intermolecular forces act at a finite distance (albeit small). They are then accelerated (still approaching the train from a tiny but finite distance) until the speeds match.Edited by sinequanon, : No reason given.

Perhaps. But the way I put it doesn't make the head spin.

I could give you another series where it 'seems implied that I do not think it diverges', and for which your logic would fail.I suppose what you are saying is that it is 'obvious' to you that this one converges, so you didn't bother to prove it?

It wasn't just incomplete, it was a wrong method. Convergence would normally be proved by showing the series tends to one. In this case assuming convergence would be assuming the thing you are trying to prove.

Take Agobot's post#13. He has implied the series diverges.Your implicit use of the convergence of the series doesn't actually pinpoint the error.

Not the normal method. Much easier and more direct to write down the sum to n terms of a geometric series and examine its difference with 1 as a function of n.

Values of limits of great interest in "real mathematics" papers. Every time you solve a differential equation or an integral, you are evaluating a limit.Applying the formulae you mentioned would not be taken as proof in a "real mathematics" exam. More like a demonstration.Have you ever had the displeasure of marking maths undergraduate exam papers?

Me too. Majored at Cambridge University here in the UK and went on to research in non-deterministic fluid dynamic modelling of mid-latitude weather systems.Would have got a fail in my first year exams if I'd submitted a proof depending on the two formulae you supplied. You'd be expected to do it more from first principles.

The sweeping wonders of set theory, eh? It allows you to talk of things for which no representation exists. Then the axiom of choice allows you to select elements from this 'soup' of abstraction.