Re: I have sinned
- From: Robert Israel <israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 05 Nov 2008 18:58:39 -0600
Many years ago when I could plot y=sin(x) on my home computer, say from
-2pi
to +2pi, I got the expected curve shape. Then just out of curiosity I added
another sin and plotted y=sin(sin(x)) and I noticed that the curve became
more rounded. So I added more sin. y=sin(sin(sin(sin(sin(.... etc.
Does this have some kind of limit? What happens to the curve if you keep
adding sin?
Just curious. It doesn't matter if there's no simple answer.
Yes, it has a limit, namely 0.
Let f_0(x) = x and f_{k+1}(x) = sin(f_k(x)).
In particular, |f_1(x)| = |sin(x)| <= 1.
Note that if 0 <= x <= 1, 0 <= sin(x) <= x <= 1
while if -1 <= x <= 0, -1 <= x <= sin(x) <= 0.
So the sequence {f_k(x)} becomes monotonic after k=1.
A bounded monotonic sequence has a limit.
Now if the limit is y = f(x), you'd have f(x) = sin(f(x)).
But the only real solution to t = sin(t) is t=0.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
- Follow-Ups:
- Re: I have sinned
- From: Zdislav V. Kovarik
- Re: I have sinned
- From: Billiam
- Re: I have sinned
- From: Old Guy
- Re: I have sinned
- References:
- I have sinned
- From: Old Guy
- I have sinned
- Prev by Date: Re: Set with cofinite topology is compact
- Next by Date: Re: JSH: Just consider Andrew Wiles
- Previous by thread: I have sinned
- Next by thread: Re: I have sinned
- Index(es):
Relevant Pages
|
