From: SourceForge.net <no...@so...> - 2008-08-30 18:49:55
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Bugs item #2083561, was opened at 2008-08-30 02:28 Message generated for change (Comment added) made by dgildea You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2083561&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: None Group: None >Status: Closed >Resolution: Fixed Priority: 5 Private: No Submitted By: Andrej Vodopivec (andrejv) >Assigned to: Dan Gildea (dgildea) Summary: Limit of the Wallis product Initial Comment: 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 ---------------------------------------------------------------------- >Comment By: Dan Gildea (dgildea) Date: 2008-08-30 14:49 Message: Logged In: YES user_id=1797506 Originator: NO 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 ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2083561&group_id=4933 |