Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

On Wed, 23 Jan 2008 22:47:52 -0800 (PST), lwalke3@xxxxxxxxx wrote:

On Jan 23, 8:52 am, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:
Conclusion: F(N) is the empty set, F(N) = {} . And:
An infinite composition of bijections is not nececcarily a
bijection.
An infinite composition of functions is not defined at all.  

So I notice how there seems to be some sort of problem with
the term "infinite composition of functions (or permutations)."

Interestingly enough, there's another recent thread here in this
newsgroup, called "Puzzle With Functions," which also deals
with infinite compositions of functions -- in this case, whether
there exists an infinite sequence of polynomial functions whose
composition is the exponential function. There was a little
cotroversy in that thread about what exactly an infinite function
composition is, but certainly nowhere near as much as here.

Perhaps because the participants in that thread were not
as confused as the ones here?

In other threads, one often states that there is no such thing
as an infinite _sum_ -- there can only be infinite _series_. If
this were algebra/ring theory, then of course all sums are
required to be finite, but in analysis, I believe that the
distinction between a finite "sum" and an infinite "series" is a
bit pedantic. We still use infinitely many "+" signs, or even a
summation sign (sigma), to denote infinite series, so as long
as the series converges, I see no reason to say that an
infinite series is not an infinite "sum."

And it's especially pedantic to state that there can be only
finite compositions of functions -- especially since there's no
convenient term like "series" for compositions. As far as I'm
concerned, infinite compositions -- once again, as long as
certain convergence requirements are met -- really do exist.

_as long as certain convergence requirements are met_ fine,
if you want to talk about the composition of infinitely
many functions instead of a limit of finite compositions
that's just great - yes, _if_ we're careful about those
convergence requirements then insisting that one say
this instead of that is somewhat pedantic.

But it seems like you haven't been paying attention.
The problem is that some people here insist on talking
about infinite compositions _without_ worrying about
convergence.

Certainly no one has any problem with an infinite composition
of functions if all but finitely many of them are the identity
function -- just as no one has any problem with an infinite
series with only finitely many nonzero terms.

So I agree with HdB that we can define f(x) to be the
composition of infinitely many permutations. I disagree
with HdB when he insists that f is itself a permutation. And
of course, one can't take the composition of infinitely many
copies of f, as it fails to converge using any reasonable
definition of convergence.

Then I don't see what your point is.

(Recall that in analysis we have pointwise convergence,
uniform converengence, L^2, etc. but here we're dealing
with natural numbers, so the only sequences that converge
are those which are eventually constant.)

.

Relevant Pages

  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of functions is not ... distinction between a finite "sum" and an infinite ... i have been called an idiot for saying sum instead ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of functions is not ... even if riemann series theorem does not apply, ... So sure enough you _don't_ know what a summability method ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of bijections is not nececcarily a ... the term "infinite composition of functions." ... certain convergence requirements are met -- really do exist. ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of functions is not ... i have been called an idiot for saying sum instead ... certain convergence requirements are met -- really ...
    (sci.math)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... An infinite composition of functions is not defined ... i have been called an idiot for saying sum instead of series or vice versa. ... certain convergence requirements are met -- really do ...
    (sci.math)