Re: -- A sequence or real numbers

cosv@xxxxxxxxxxxxxxx wrote:
Does anyone know, or at least have a clue on the following question ?

Go on, I'll bite, although it's hardly research [having said that,
at least the question has a meaning, unlike that apparently endless
logic thread, which appears to be the first 3 lectures of Model Theory 101
plus a lot of noise...]

Let (b_n)_n be the sequence defined by :

b_1=1/4 , b_2=1/16 and

b_{n+1}=b_1 b_n+b_2 b_{n-1}+ ... +b_k b_{n-k+1}+ ... +b_1 b_n

Use generating functions! If F is sum b_n X^n then one checks
easily that F^2=F-X/4.

Now solve the quadratic to deduce that 1-2F is the power series (1-X)^(1/2).

Now use the binomial theorem to get an explicit formula for the b_n,
involving things like the product of the first (n-1) odd integers,
which you probably spotted if you computed a few b_n exactly.

Now write everything in terms of factorials and deduce

b_n=(2*n-2)!/2^(2*n)/(n!)/((n-1)!)

Now use Stirlings formula to deduce what you want [modulo monotonicity,
which follows from elementary arguments involving
looking at the ratio of b_n to b_{n+1} and the concavity of log].

A computer evaluation hints to the fact that (X_n) is decreasing to a
limit approx equal to 1.141...

This proof shows that X_n tends to a limit exactly equal to the
reciprocal of 4.sqrt(pi).

Kevin (ten minutes of my life I'll never get back!)

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