Prove that a polynomial of this form has a real root of this form.

Question

Prove or disprove that for positive coefficients:

$$a_k>0$$

this polynomial:

$$a_1x^0+a_2x^1+a_3x^2+a_4x^3+...+a_nx^{n-1}-x^n=0$$

has a real root $x$ that is the fraction from the following linearly recursive sequence:

$$t(0)=1,\;\;t(1)=1,\;\;t(2)=1,\;\;t(3)=1,\;...\;,t(n)=1$$ $$t(n)=\sum_{k=1}^{k=n} a_{n-(k-1)}*t(m - k)$$

where $$m=\text{degree of polynomial}$$

$$x=\lim_{n \rightarrow \infty}\frac{t(n)}{t(n-1)}$$

Please refer to this Mathematica 8 program:

Clear[t, n, nn, x]
nnn=5;
a = Table[RandomReal[WorkingPrecision -> 50], {n, 1, nnn}];
Print["polynomial:"]
Total[Flatten[{a*Table[x^(n - 1), {n, 1, nnn}], -1*x^nnn}]]
Print["polynomial degree:"]
nn = Length[a]
mm = 1000;
t[0] = 1;
t[1] = 1;
t[2] = 1;
t[3] = 1;
t[4] = 1;
t[n_] := t[n] = 
   a[[5 - (5 - 1)]]*t[n - 5] + a[[5 - (4 - 1)]]*t[n - 4] + 
    a[[5 - (3 - 1)]]*t[n - 3] + a[[5 - (2 - 1)]]*t[n - 2] + 
    a[[5 - (1 - 1)]]*t[n - 1];
Print["If the following two numbers are equal then the ratio has \
probably converged:"]
N[t[mm - 1]/t[mm - 2], 50]
x = N[t[mm]/t[mm - 1], 50]
Print["If this number is zero then x is a root of the polynomial \
above"]
Sum[a[[n]]*x^(n - 1), {n, 1, nn}] - x^nn

in case the latex is incorrect for the sum in the recurrence. The program shows the solution for the quintic case.

Also if you can make the restrictions on the coefficients more precise (broader) than this I would be happy.