[math-fun] Sequence of polynomials with largest real root approaching e?
Mike Stay wants polynomials p_n(x) of degree n so that 1. p_n(x) is degree n 2. p_{n+1}(x) = x * p_n(x) + k_n, where k_n is an integer 3. Largest real roots approach e as n -> infinity. --I argue that such polynomials exist, not only for e=2.71828..., but in fact for ANY desired real number e>1. Clearly, if e=A^(1/B) for any integers A,B>0 one can hit it exactly using the polynomial x^B - A. Then all subsequent k_n=0 so you stay there forever after. These numbers seem obviously dense within the reals>=1 (one can in fact prove that by showing within any subinterval one exists...) so you clearly can approximate any e>1 as closely as you like. However that is not quite good enough to satisfy Stay's desire. But let us say you have found some A,B>0 giving you accuracy 10 decimals and yielding a number slightly BELOW e. Now you can find C,D>0 so that x^C * (x^B-A) - D gives you accuracy 100 decimals and still below e. Then find E,F>0 so x^E * [x^C * (x^B-A) - D] - F gives you 1000 decimals... still below e... etc. I claim it is fairly obvious that you can keep doing this forever (a full proof would need various details here... which I omit since I am lazy...). So the answer to Stay's question is "yes." -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Thanks! The reason I ask is I'd like something like base-Fibonacci where certain patterns are excluded but where the place values grow like e^n instead of phi^n. On Tue, Apr 16, 2013 at 10:34 AM, Warren D Smith <warren.wds@gmail.com> wrote:
Mike Stay wants polynomials p_n(x) of degree n so that 1. p_n(x) is degree n 2. p_{n+1}(x) = x * p_n(x) + k_n, where k_n is an integer 3. Largest real roots approach e as n -> infinity.
--I argue that such polynomials exist, not only for e=2.71828..., but in fact for ANY desired real number e>1. Clearly, if e=A^(1/B) for any integers A,B>0 one can hit it exactly using the polynomial x^B - A. Then all subsequent k_n=0 so you stay there forever after. These numbers seem obviously dense within the reals>=1 (one can in fact prove that by showing within any subinterval one exists...) so you clearly can approximate any e>1 as closely as you like. However that is not quite good enough to satisfy Stay's desire.
But let us say you have found some A,B>0 giving you accuracy 10 decimals and yielding a number slightly BELOW e. Now you can find C,D>0 so that x^C * (x^B-A) - D gives you accuracy 100 decimals and still below e. Then find E,F>0 so x^E * [x^C * (x^B-A) - D] - F gives you 1000 decimals... still below e... etc. I claim it is fairly obvious that you can keep doing this forever (a full proof would need various details here... which I omit since I am lazy...).
So the answer to Stay's question is "yes."
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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Warren D Smith