Sqrt of 2 by Newtons method FullSimplify[ ]-ied

I got this formula for the square root of 2 published in the OEIS:
sqrt(2) = lim_{n->infinity}(1/(1-Sum_{k=1..n}(((-1)^(k-1)*binomial(n-1, k-1))/f(k/n+s-1/n))/Sum_{k=1..n}(((-1)^(k-1)*binomial(n-1, k-1))/f(k/n + s)))+s)
where f(x)=x^2-2 and s=1.
=1.41421356237309504880168872420969807857...

(* Mathematica start *)
Clear[f,s,n,k];
f[x_] := x^2 - 2;
s = 0;
n = 1000;
N[(1/(1 -
      Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/f[k/n + s - 1/n], {k,
         1, n}]/Sum[((-1)^(k - 1)*Binomial[n - 1, k - 1])/
         f[k/n + s], {k, 1, n}]) + s), 40]
f[%]
(* Mathematica end *)

The convergence
depends on the choice of the seed point "s=0" and the number of iterations "n=1000".
f[x_] := x^2 - 2 is the function whose root is searched for.

It would be interesting to know which functions are solvable
by this convergent or nonconvergent ratio for the choice of seed point "s=0".