Re: [math-fun] mis-defining limits
Richard Guy asks: << Is there any difficulty as x --> 0- or 0+ ? I thought that the interesting question is x --> 1- R. . . . On 9/10/06, Daniel Asimov <dasimov@earthlink.net> wrote: < Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . . (which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
Good point, Richard, and the limit as x -> 0 is not that interesting, either. I posted the wrong question, yet another mistake created by undue haste. *****The question of interest is: "What happens to g(x) as x -> 1- ?" --Dan
Indeed, I conjecture that g(z) (as defined below) and all its derivatives converge at every point of the unit circle in the complex plane; then writing z = exp(i t) gives a periodic function of real variable t, whose real part h(t) = Re(g(exp(i t))) is given by a Fourier series which will no doubt be Cesaro summable. This function might be a simpler example of the type I mentioned earlier; but it may well have discontinuities, as do the standard Fourier series examples --- square wave, sawtooth, etc. Can somebody whose analysis is less rusty than mine cast some light here? WFL On 9/11/06, Daniel Asimov <dasimov@earthlink.net> wrote:
Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . .
(which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
Good point, Richard, and the limit as x -> 0 is not that interesting, either.
I posted the wrong question, yet another mistake created by undue haste.
*****The question of interest is: "What happens to g(x) as x -> 1- ?"
--Dan
On 9/11/06, Daniel Asimov <dasimov@earthlink.net> wrote:
Btw, on his website, Noam Elikies asks what the limit is of the infinite series g(x) = x - x^2 + x^4 - x^8 + . . . +- x^(2^k) -+ . . .
(which converges for |x| < 1) as x -> 0-. It's a cool problem to let students try to guess the answer just using calculators.
Good point, Richard, and the limit as x -> 0 is not that interesting,
either.
I posted the wrong question, yet another mistake created by undue
haste.
*****The question of interest is: "What happens to g(x) as x -> 1-
?"
--Dan
Plot g(1-exp(-x)). For x > 6, the graph becomes periodic. g(x) ~ 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385). Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com
On 9/12/06, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Plot g(1-exp(-x)). For x > 6, the graph becomes periodic.
g(x) ~ 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385).
Gene
I think maybe the above should have read g(1-exp(-x)) = 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385) which does look to be true as x -> 1- with error O(1-x). All that spiel about g(x) converging on the unit circle is codswallop, because the relation g(x) + g(x^2) = x fails to hold in the limit --- nice one! If I'd listened to advice and "used a calulator" in the first place, I might have spared myself a very red face here. WFL
1.385 looks suspiciously like 2 log 2 = 1.3862... Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Fred lunnon Sent: Tue 9/12/2006 10:08 AM To: math-fun Subject: Re: [math-fun] mis-defining limits On 9/12/06, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Plot g(1-exp(-x)). For x > 6, the graph becomes periodic.
g(x) ~ 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385).
Gene
I think maybe the above should have read g(1-exp(-x)) = 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385) which does look to be true as x -> 1- with error O(1-x). All that spiel about g(x) converging on the unit circle is codswallop, because the relation g(x) + g(x^2) = x fails to hold in the limit --- nice one! If I'd listened to advice and "used a calulator" in the first place, I might have spared myself a very red face here. WFL _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Using roots of g(1-exp(-x)) near x=30, the period is 1.38629436093 vs. 2 log(2) = 1.38629436111. --- "Schroeppel, Richard" <rschroe@sandia.gov> wrote:
1.385 looks suspiciously like 2 log 2 = 1.3862...
Rich
-----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Fred lunnon Sent: Tue 9/12/2006 10:08 AM To: math-fun Subject: Re: [math-fun] mis-defining limits
On 9/12/06, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Plot g(1-exp(-x)). For x > 6, the graph becomes periodic.
g(x) ~ 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385).
Gene
I think maybe the above should have read
g(1-exp(-x)) = 0.5 + 0.027 sin(2 pi (x - 0.526) /1.385)
which does look to be true as x -> 1- with error O(1-x).
All that spiel about g(x) converging on the unit circle is codswallop, because the relation g(x) + g(x^2) = x fails to hold in the limit --- nice one!
If I'd listened to advice and "used a calulator" in the first place, I might have spared myself a very red face here.
WFL
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participants (4)
-
Daniel Asimov -
Eugene Salamin -
Fred lunnon -
Schroeppel, Richard