-----Original Message----- On Behalf Of R. William Gosper Sent: Friday, January 30, 2004 3:10 AM To: math-fun@mailman.xmission.com; staym@clear.net.nz Subject: Re: [math-fun] limit question
Simon> C(n) is slightly lower than 4^n, while the principal term
is 4^n, eventually the sum converge to 2, I am not a combinatorist or a wilf-zeilbergerologist but I can say that there is probably a variety of ways to show that.
My HYPERSIMP Macsyma function can sum Catalan(n)/4^n (without WZ): (c29) (sum(binomial(2*n,n)/(n+1)/4^n,n,0,inf),%% = hypersimp(%%))
inf ==== \ binomial(2 n, n) (d29) > ---------------- = 2 / n ==== 4 (n + 1) n = 0
Maple can do that sum, too, sum(binomial(2*n,n)/4^n/(n+1),n=0..infinity); 2 This is a simple thing. The generating function for Catalan numbers is infinity 1/2 ----- 1 - (1 - 4 x) \ n ---------------- = ) C(n) x 2 x / ----- n = 0 For x=1/4, it equals 2. The series is convergent because C(n) ~ 4^n/(pi^(1/2)*n^(3/2)). Alec Mihailovs http://webpages.shepherd.edu/amihailo/