Quote:Original post by CloudNine
First is, is the n-body problem, for n > 2, actually solvable in an equation? I've heard it's been proven to be impossible, but that does seem a bit limiting. We can put men on the moon, but we can't solve something like that?

It can't be solved analytically.
This means that you can't find a function (which is a composition of elementary functions) that would describe the motion. The problem can be solved by numerical means of course. Any physical simulator does just that. :)

A well known analytic solution exists for the 2-body problem which is closely related to Kepler's laws of planetary motion.

Quote:
Second, what is a Cauchy sequence? I've seen the wikipedia and wikibook articles, but they don't seem to give many examples. Would the sum of (1/2r) from r=0 to infinity (which has a limit at 2) be a Cauchy sequence?

Given a sequence (an), it is said to be a Cauchy sequence iff for any e>0 there is N>0 such that for any n,m>N we have |an - am| < e.
In Euclidian space, a sequence is a Cauchy sequence iff its converging.
By the way, the series (1/2r) doesn't converge.

[Edited by - ury on February 10, 2006 5:29:13 AM]

The series: sum[1/r2, r=1..inf] converges. In fact, it can be proven that sum[1/ra, r=1..inf] converges for any a > 1.

Given series: sum[ai, i=1..inf], we can construct a sequence like this: Sn=sum[ai, i=1..n].
The series converges iff the sequence Sn converges.

Since the series in your question converges, the appropriate Sn converges as well, thus Sn is a Cauchy sequence.

Indeed, real numbers can be constructed from rational numbers using Cauchy sequences.

This is theoretical, but it's the actual definition...
If, no-matter how small the epsilon you choose, (some positive real number), there is always an index n0 (natural number), such that the distance of any two terms of the sequence -above n0- is smaller than epsilon, then the sequence is said to be a Cauchy sequence, and this guarantees that it also converges to some real number. This must be true for all positive epsilon.

E.g. the sequence an = 1/n is a Cauchy sequence, because for any epsilon, e.g. e=e0>0, you can find an index n0, such that for all m,n > n0 the following inequality should hold: |an - am| < e0

|an - am| = |1/n - 1/m| < |1/n0| + |1/n0| = 2/n0
Therefore: 2/n0 < e0 =>
n0 > 2/e0

The sum of 1/2r for varying r, that you mention, is not a sequence, it is a series, the series of partial sums of the sequence 1/2n.

edit:
trivial

[Edited by - someusername on February 12, 2006 4:38:08 PM]