A Generalization of the Sequence Convergence Problem

[Pe Joseph-AJP070]

Here is a generalization of the sequence convergence problem I posted
yesterday. (For convenience, I append
the original problem at the end of this message.)

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Let s(n) be a sequence that converges to a real number K different from -1.
Define the "oscillator sequence" a(n) of s(n) by the rules:

                     a(1) = 1;
                     a(n) = 1 - (s(n-1)/s(n)) a(n-1) if n > 1.

Note that the original problem below concerns the convergence of the
oscillator sequence of s(n) = p(n).  

Examples of s(n) are s(n) = n, s(n) = n^2 and in general, any polynomial
function in n that is not the zero polyonomial function. 
The oscillator sequence of s(n) may converge or diverge depending on s(n).
Clearly, if s(n) = 1 is the constant sequence mapping 
each positive integer to 1, then a(n) diverges--in fact, oscillates between
0 and 1 (hence the name "oscillator sequence"). 
If s(n) = n, it is not hard to prove that a(n) converges to 1/2.

Can you find conditions on s(n) that will ensure the convergence of its
oscillator sequence? Of course, if s(n) converges, then 
lim s(n) = 1/(K + 1).

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J. L. Pe

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Original Problem:

Define the sequence a(n) by: a(1) = 1; a(n) = 1-(p(n-1)/p(n))*a(n-1) if n >
1, where p(n) denotes the n-th prime. 
It's easy to show (an exercise!) that if L = lim a(n) exists, then L = 1/2. 
Can you prove the convergence of a(n) or the divergence of a(n)?