Wikipedia:Reference desk/Archives/Mathematics/2010 September 6

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September 6

Proving the sequence defining e converges

How would you go about proving that $\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$ converges? Or, say, that $2<\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}<3$, then that $2.70<\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}<2.72$ and so on, zooming in as far as you like? It Is Me Here ^{t / c} 17:54, 6 September 2010 (UTC)

You can use the binomial theorem, $\left(1+{\frac {1}{n}}\right)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {1}{n^{k}}}$. It's easy to show that the kth term is less than $1/k!$ so the expression is less than $\sum _{k=0}^{\infty }{\frac {1}{k!}}$. In fact the kth term converges to $1/k!$, and it should be possible to bound the differences to make the entire expression converges.

There should be a result that states conditions under which $\lim _{n\to \infty }\sum _{k=0}^{\infty }a_{n,k}=\sum _{k=0}^{\infty }\lim _{n\to \infty }a_{n,k}$, but I can't remember or find the details. -- Meni Rosenfeld (talk) 18:17, 6 September 2010 (UTC)

The dominated convergence theorem gives convergence here. There's probably a separate name for dominated convergence applied to discrete measure spaces, but if so I don't know it. Algebraist 18:28, 6 September 2010 (UTC)

Also, you can show that the sequence $(1+1/n)^{n}$ is (strictly) increasing: use the inequality of arithmetic and geometric means with choice of $n+1$ numbers $x_{1}=x_{2}=\dots =x_{n}:=1+1/n$ and $\textstyle x_{n+1}:=1.$ To show that it is bounded, note that the argument above also holds more in general for the sequence $(1+c/n)^{n}$ for any real $c$, showing it is increasing as soon as $n+c>0.$ Since for n>|c|

$\left(1+{\frac {c}{n}}\right)^{n}\left(1-{\frac {c}{n}}\right)^{n}=\left(1-{\frac {c^{2}}{n^{2}}}\right)^{n}\leq 1$

and both sequences on the LHS are positive and increasing, they are bounded. So this actually gives the convergence of $(1+c/n)^{n}$ for all real $c$ together with the bounds: in particular for $c=1$ you should obtain, for all $n$

$\left(1+{\frac {1}{n}}\right)^{n}<e<\left(1+{\frac {1}{n}}\right)^{n+1}.$

Note however that the argument via the dominated convergence theorem for series shown by Algebraist also holds when $c$ is more generally a complex number, or a matrix, or even an element in a Banach algebra. Also, it can be shown that the convergence of the series is much faster. --pma 02:46, 7 September 2010 (UTC)

The Space of Oriented Geodesics

Consider the a 2-sphere is 3-space, centred at the origin. The geodesics are given by the great circles. For each great circle we get a unique plane, namely the plane containing the great circle (and passing through the origin). Given a plane passing through the origin we get a unique line passing through the origin, namely the line perpendicular to the given plane passing through the origin. So each great circle gives a line through the origin. The same procedure works in reverse too: a line through the origin gives a plane through the origin (the plane passing through the origin that is perpendicular to the line). A plane through the origin gives a great circle (the circle generated by intersecting the sphere with the plane). It follows that the space of unoriented geodesics is exactly RP², i.e. the real projective plane. Now consider the space of oriented geodesics of the sphere. This is exactly the sphere itself. Given a point p of the sphere, we get an unoriented geodesic by intersecting the sphere with the plane passing through the origin that is perpendicular to the chord joining the origin to p. Let us denote this intersection by C. Assume that g : S¹ → S² gives a smooth parametrisation of C, with parameter θ. We can give an orientation to C by insisting that

${\text{Volume}}\left\{\left[\,p,\,g(\theta ),\,g'(\theta )\,\right]\right\}>0\ {\text{ for all }}\ \theta .$

Clearly, if we replace p by −p then we must reverse the orientation of C, i.e. replace θ with –θ, in order to maintain the inequality above. This means that there is a one-to-one correspondence between the oriented geodesics of the 2-sphere and the 2-sphere itself. My question is this:

• What other manifolds M have the property that there is a one-to-one correspondence between the oriented geodesics of M and M itself?

— Fly by Night (talk) 18:14, 6 September 2010 (UTC)

If two sets have the same cardinality, then there is a one-to-one correspondence between them. The cardinality of the set of real numbers is c. The cardinality of the set of points in a manifold is also c. The cardinality of the set of oriented geodesics in a manifold is also c. So all manifolds M have the property that there is a one-to-one correspondence between the oriented geodesics of M and M itself. You did not require the correspondance to be continuous. Bo Jacoby (talk) 23:37, 6 September 2010 (UTC).

Could you please show how you prove that the space of oriented geodesics on M has the same cardinality as M? — Fly by Night (talk) 00:57, 7 September 2010 (UTC)

The set of all points in an infinite plane has the same cardinality as the set of all points in a finite line segment, namely, c.[1] What is the one-to-one correspondence between an infinite plane and a finite line segment? — Fly by Night (talk) 00:10, 7 September 2010 (UTC)

Well, finding a specific one is a little complicated. Not too bad, but not something you can draw on a napkin like the one between the rationals and the natural numbers.

The easiest approach to giving an explicit bijection is probably to find injections both ways, and then apply the method used to prove the Schröder–Bernstein theorem. Finding an injection from the line segment to the plane is trivial; you can get the reverse injection by something such as interleaving the decimal representations of the x- and y- coordinates of a point, after projecting the plane into a finite square. --Trovatore (talk) 00:39, 7 September 2010 (UTC)

Thanks Trovatore. Do you think this cardinality argument shows that there is a bijective correspondence between the space of oriented geodesics on M and M itself? I mean, how can Bo say that the space of oriented geodesics on M has the same cardinality as M? Geodesics are differential invariants, whereas cardinality is much weaker. Making small bumps on the sphere may stop is having any closed geodesics at all. — Fly by Night (talk) 00:46, 7 September 2010 (UTC)

N.B. I meant to say that there is a bijective correspondence between the 2-sphere and its space of oriented geodesics. And my question should have been: What other manifolds M have the property that there is a bijective correspondence between the oriented geodesics of M and M itself? — Fly by Night (talk) 00:10, 7 September 2010 (UTC)

"how can Bo say that the space of oriented geodesics on M has the same cardinality as M?" An oriented geodesic is defined by the two points (P, P+dP). The cardinality of pairs of points is c when the cardinality of points is c. Bo Jacoby (talk) 06:25, 7 September 2010 (UTC).

Why not try a stronger condition that just bijection, such as Homeomorphism, or Diffeomorphism both of which are exhibited by the sphere? This leads to a more general question of when we can give the space of all geodesics a manifold structure?--Salix (talk): 09:33, 7 September 2010 (UTC)

It seems like there is a bit of existing work on the space of geodesics [2] has a survey article. There are several results about the topological structure of the space, including conditions for it to be a manifold, but falls short of answering Fly by Night's question.--Salix (talk): 10:07, 7 September 2010 (UTC)

That's really interesting. Thanks Salix. This is more like what I had in mind, i.e. does the space of oriented geodesics have a manifold structure and is it the same as the original manifold. I know I didn't say as much but I think you hit the nail on the head. Thanks again! — Fly by Night (talk) 16:41, 7 September 2010 (UTC)

big-O?

I've got some calc homework and I need to solve a problem using the big-O. Unfortunately I forgot how (it's been a long labor day weekend ;) and your article is kind of written more for the person looking something up that they'd learnt years ago than me, just learning it now. I'm not going to give the actual problem because it would be kind of pointless if I didn't figure out how to do it myself, but can someone show me how in simple terms (using any problem)? Thanks 76.229.163.32 (talk) 19:59, 6 September 2010 (UTC)

If you won't give an explicit problem then you must be seeking a general solution. In that case; you'll find everything you need in the formal definition section. — Fly by Night (talk) 20:43, 6 September 2010 (UTC)