#1484 Limit of the Wallis product

[Andrej Vodopivec]

Computing the Wallis product for %pi fails:

(%o3) (%pi*4^N*N!^2)/(2*2^(2*N)*gamma(N+1/2)*gamma(N+3/2))
(%i4) limit(%, N, inf);
(%o4) 0
(%i5) load(stirling)$
(%i6) stirling(%o3);
(%o6) ((N+3/2)^(-N-1)*2^(-2*N-1)*4^N*%e^(2*N+2)*N!^2)/(2*(N+1/2)^N)
(%i7) limit(%, N, inf);
(%o7) 0
(%i8) ratsimp(%o6);
(%o8) (4^N*%e^(2*N+2)*N!^2)/((2*N+1)^N*(2*N+3)^N*(4*N+6))
(%i9) limit(%, N, inf);
(%o9) %pi/2

Only the last limit is correct.

Andrej

[Dan Gildea]

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src/tlimit.lisp rev 1.7:
taylim: ask for $lhospitallim terms of taylor series, instead of 1 term.
this is an arbitrary limit: with default value $lhospitallim = 4,
tlimit(2^n/n^5, n, inf) => 0
(before, tlimit(2^n/n^2, n, inf) => 0 )
This handles this problem with the default settings, and gives the
user the ability to increase the limit.

(%i2) (%pi*4^N*N!^2)/(2*2^(2*N)*gamma(N+1/2)*gamma(N+3/2));
(%o2) %pi*2^(-2*N-1)*4^N*N!^2/(gamma(N+1/2)*gamma(N+3/2))
(%i3) limit(%, N, inf);
(%o3) %pi/2
(%i4) load(stirling)$
(%i5) stirling(%o2);
(%o5) (N+3/2)^(-N-1)*2^(-2*N-2)*4^N*%e^(2*N+2)*N!^2/(N+1/2)^N
(%i6) limit(%, N, inf);
(%o6) %pi/2
(%i7) ratsimp(%o5);
(%o7) 4^N*%e^(2*N+2)*N!^2/((2*N+1)^N*(2*N+3)^N*(4*N+6))
(%i8) limit(%, N, inf);
(%o8) %pi/2