I’d like to solve the Limit of the following Sequence on Mathematica 12 :
DiscreteLimit[(1 + Sin[Pi*Sqrt[4*n^2 + 1]])^n, n -> Infinity]
No results given by Mathematica.
But I do some corresponding equivalent deformation,
Sin[Pi*Sqrt[4*n^2 + 1]=Sin[-2*n*Pi+Pi*Sqrt[4*n^2 + 1]
=Sin[Pi/(Sqrt[4*n^2 + 1] + 2*n)]
DiscreteLimit[(1 + Sin[Pi/(Sqrt[4*n^2 + 1] + 2*n)])^n, n -> Infinity]
I got the corresponding results
fortunately： E^([Pi]/4).
What is the effect of the form of the expression on seeking the limit？
Which direction should I try to get the result? I want to know some reason for this.