If a series converges at infinity, then it can be determined, which is the whole point. And we already know that you cannot know the “exact value” for pi, at least in decimal digits. It’s a transcendental number.
If there are two points, A(x1, y1, z1) and B(x2, y2, z2), where x1 != x2 and/or y1 != y2 and/or z1 != z2, there is some distance between them which can be arbitrarily small.
You’re talking about Zeno’s Paradox. The series of reciprocals of powers of two is convergent, and actually equals 2 at infinity.
Yes, one of them. But hey! That’s not fair! It would seem to introduce time into the calculations.
Now I have to stand corrected, errors in calculations do not just apply to very large distances, but also very small ones. Mini-Macro sort of thing.
Not sure that this would be exactly on topic, but have you read about how the calculations for what may happen with the firing up and experiments of the LHC have been revisited and now questions are being asked… http://arxivblog.com/?p=1136
Thanks for the link on convergent series, it may take a while for me to wade through that.
These stories are frustrating because the scientists are being truthful when they say that the chance exists that something very bad could happen, but in order to communicate to lay audiences, they leave out figures, which allows the public to go wild with imagination. As long as the probability is not exactly zero, we must say there is some chance.
What people don’t realize is that the Earth is constantly bombarded by particles similar to or more energetic than those of the LHC. The LHC will smash protons together with a maximum energy of 7x10^12 eV. The Earth is hit by particles with energies of 10^20 eV and greater. That is to say, even with the LHC pumped up to maximum power, we are still hit by particles from space with 100,000,000 times as much energy. And this has being going on for ~4.5 billion years (as long as the Earth has been around), and a menacing black hole has yet to be produced. So when the scientists say the chance of such an event happening is small, they know what they are talking about.
I have lost my touch with Maths as I used to have it back in high school (I went studying languages, Finnish to be precise), but this book is nonetheless very, very readable and understandable. And also very enjoyable.