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I need to evaluate the limit:

$$\lim_{n\to\infty}\prod_{k=1}^\infty \left(1-\frac{n}{\left(\frac{n+\sqrt{n^2+4}}{2}\right)^k+\frac{n+\sqrt{n^2+4}}{2}}\right).$$

I could not type into WolframAlpha and find its value. Can someone help me?

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    $\begingroup$ The way you could type that into WolframAlpha, or Mathematica is Limit[Product[1-n/(((n+Sqrt[n^2+4])/2)^k+(n+Sqrt[n^2+4])/2),{k,1,Infinity}],n->Infinity] but it appears that neither of those are able to see a way to give you the limit. They do not appear to solve even the simpler Limit[Product[1-n/(n^k+n),{k,1,Infinity}],n->Infinity] Perhaps you can think of a way to simplify your problem. $\endgroup$ – Bill Aug 29 at 18:46
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With Mathematica I have:

func[n_] := NProduct[1 - n/(((n + Sqrt[n^2 + 4])/2)^k + (n + Sqrt[n^2 + 4])/2), {k, 1, Infinity}]
Table[func[10^n], {n, 1, 10}]

(*{0.454951, 0.49505, 0.4995, 0.49995, 0.499995, 0.5, 0.5, 0.5, 0.5, 0.5} *)

It looks like the limit is: 1/2

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Analytically, it is easy to see that the first term in the product has a limiting value of 1/2 and all the other terms have limiting values of 1. My maths is a bit rusty, but I suspect that you can justify changing the order of the two limiting processes.

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