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?

  • 1
    $\begingroup$ 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

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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.