This entry was posted
on Tuesday, October 3rd, 2006 at 11:02 am and is filed under General.
You can follow any responses to this entry through the RSS 2.0 feed.
You can skip to the end and leave a response. Pinging is currently not allowed.
If you don’t mind recklessly throwing all caution to the wind here’s another way to ‘derive’ the Taylor series, different to my previous ‘derivation’:
If F[X] is an F-container of X’s
dF[X] is an F-container with a hole
d^2F[X} is an F-container with an ordered pair of holes
(1+d^2/2)F[X} is an F-container with no holes or an unordered pair of holes, etc.
G[d]F[X] is an F-container with a G-container of holes in it(!!)
A.dF[X] is an F-container with an A-labelled hole
G[A.d]F[X] is an F-container with a G-container of A-labelled holes in it
exp(A.d)F[X] is an F-container with a set of A-labelled holes in it
But that’s just an F[X+A].
Expanding the left hand side gives the Taylor series.
But yes, I’m fully aware this is a long way from rigour. But it’s fun to play with, and a container of holes sounds like a useful thing to me. I think it’d be nice to make it rigorous somehow.