Limit of a product doesn't work

Questions
1

Hi everyone,
I’m new with Mathematica and i don’t understand why this instruction doesn’t work:

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

Can anyone help me please?

2

Really a puzzle

k (n - k )/n^4 // Expand

-(k^2/n^4)+k/n^3

SetAttributes[f, {Listable, NumericFunction}]

f[n_] :=   With[{n1 = Floor[n]},    Product[1 + Sqrt[k/n1 (1 - k/n1)]/n1, {k, 1, n1}]]

In=  Limit[f[n], n -> \[Infinity]]

Out =  1

    With[{lim = Rationalize[f[10^6] // N, 1/100]},   
          Plot[ f[n], {n, 1, 50}, , PlotRange -> {1, 1.7},
             ImageSize -> Small,  Ticks -> {{0, 20, 40}, {0, lim }},   
             GridLines -> {None, {1, lim , 3/2}}]]

1.jpg
1.jpg300×196 4.01 KB

3

This is really a math question. Mathematica simply cannot do the limit.

First an approximation.

partials = Table[Product[1 + Sqrt[k (n - k)]/n^2, {k, 1, n}], {n, 20, 400, 20}];
N[partials]

(* Out[470]= {1.4681, 1.47549, 1.47761, 1.47858, 1.47913, 1.47948, \
1.47972, 1.4799, 1.48003, 1.48014, 1.48022, 1.48029, 1.48035, 1.4804, \
1.48044, 1.48048, 1.48051, 1.48054, 1.48057, 1.48059} *)

We can also get the limiting value numerically like so.

NLimit[Product[1 + Sqrt[k (n - k)]/n^2, {k, 1, n}], n -> Infinity]

(* Out[472]= 1.48097 *)

This happens to be a really good approximation to Exp[Pi/8]. There are a few steps required to show that that is in fact the limit.

4

Thanks to both of you.

I already proved that the limit was exp(pi/8) but I was curious about the capacities of mathematica to find it.
I didn’t know either the NLimit command which seems very useful.

Thank you very much for taking the time to answer me

5

Mathematica can certainly compute a relevant value.

Exp[Limit[Integrate[Sqrt[k (n - k)]/n^2, {k, 1, n}, 
   Assumptions -> n > 1000], n -> Infinity]]

(* Out[478]= E^(\[Pi]/8) *)

Reducing the limit to that computation is a different matter.