I have a curiosity as regards the infinite product below. I wonder why Mathematica v.8.0. says
the limit is $1$. This is not true.

Limit[(Product[(1 - a/n^2), {n, 1, Infinity}])^(1/Sqrt[a]),  a -> Infinity]

• Why do you say it is not true? – Teake Nutma Jul 21 '13 at 12:09
• @Pinguin Dirk Consider $\sqrt{a}$ is an integer. – user 1357113 Jul 21 '13 at 12:12
• V10.1 now gives the result Interval[{0, 1}]. – Michael E2 May 4 '15 at 2:18
• @MichaelE2 V12.x give 1 again. (Arguably better/worse result than Interval[{0,1}]?) – Silvia Nov 28 at 14:58
• @Silvia Probably worse in that mathematically the limit Limit[p, a -> Infinity], where p = (Sin[Sqrt[a] Pi]/(Sqrt[a] Pi))^(1/Sqrt[a]) is the product, does not exist unless one restricts the domain to Sin[Sqrt[a] Pi] != 0. OTOH, it might be considered better in that one could argue that the product "diverges" to zero when $\sqrt{a}$ is an integer, and that the domain of a should be so restricted. However, the limit is computed as 1 without such a restriction, and that is incorrect. – Michael E2 Nov 28 at 15:53

$$\prod_{n=1}^\infty\left|\,1-\frac{a}{n^2}\,\right|^{1/\sqrt{a}}$$ As $a\to\infty$, each term is like $1+\frac{\log(a)}{n^2\sqrt{a}}$ and as $\frac{\log(a)}{n^2\sqrt{a}}$ is absolutely summable to something around $\frac{\log(a)}{\sqrt{a}}\frac{\pi^2}{6}\to0$ I would say that the product limits to $1$, unless $a=n^2$ for some $n$. So the limit doesn't exist.

• did you consider the case when $\sqrt{a}$ is an integer? – user 1357113 Jul 21 '13 at 12:22
• If you take any $\sqrt{a}$ integer then the limit is precisely $0$. $$\pi ^{-\frac{1}{\sqrt{a}}} \left(\frac{\sin \left(\pi \sqrt{a}\right)}{\sqrt{a}}\right)^{\frac{1}{\sqrt{a}}}$$ – user 1357113 Jul 21 '13 at 12:29
• @Chris'ssister: and the expression you give there goes to $1$ as $a\to\infty$ – robjohn Jul 21 '13 at 12:31
• Oh, yes. one term is 0. So the limit doesn't exist. – robjohn Jul 21 '13 at 12:32
• It's nice to see you over here robjohn – Mr.Wizard Jul 21 '13 at 12:51

Any ideas why it is not true ? Typo/s ?

Running Win 8, Mathematica 9.0.1 the result is the same and also on my Ubuntu desktop with Mathematica 9.0.0 and WolframAlpha: Check here

Computing the (Product[(1 - a/n^2), {n, 1, ∞}])^(1/Sqrt[a]) yields the result:

$$\pi ^{-\frac{1}{\sqrt{a}}} \left(\frac{\sin \left(\pi \sqrt{a}\right)}{\sqrt{a}}\right)^{\frac{1}{\sqrt{a}}}$$

Taking the limit of it as $a \to \infty$ is equal to $1$

And plotting it:

Edit: Assuming $\sqrt{a}\in \mathbb{Z}$ and taking the limit of the product we end up with $0$:

Limit[(Product[(1 - a/n^2), {n, 1, ∞}])^(1/Sqrt[a]),
a -> ∞, Assumptions :> Sqrt[a] ∈ Integers]

• What if $\sqrt{a}$ is an integer? – user 1357113 Jul 21 '13 at 12:20
• @Chris'ssister How about now ? – Sektor Jul 21 '13 at 12:24
• Thanks so much! Why do you think Mathematica says the limit exists when it doesn't? – user 1357113 Jul 21 '13 at 12:26
• Technically speaking the limit still exists before we made the assumptions, but the "problem" is Mathematica was operating with Real numbers. – Sektor Jul 21 '13 at 12:34
• Glad to help :) – Sektor Jul 21 '13 at 12:37