-----Original Message----- From: Emeric Deutsch [mailto:deutsch@duke.poly.edu] Sent: Friday, January 30, 2004 7:18 AM To: alec@mihailovs.com; math-fun Subject: RE: [math-fun] limit question
Maple gives
S:=n->sum(binomial(2*k,k)/(k+1)/4^k,k=0..n): simplify(S(n));
(-n) - 1/2 binomial(2 n + 2, n + 1) 4 + 2
S(infinity);
2
I repeat, just in case, S(n)=2 - binomial(2n+2,n+1)/2^(2n+1).
That's interesting. The formula for S(n) can be also obtained using generating functions. Let S(n,x)=C(0)+C(1)x+...+C(n)x^n. The generating function for S(n,x) is infinity 1/2 ----- 1 - (1 - 4 x t) \ n ------------------ = ) S(n, x) t 2 x t (1 - t) / ----- n = 0 For x=1/4, we have infinity 1/2 ----- 2 (1 - (1 - t) ) \ n ------------------ = ) S(n) t t (1 - t) / ----- n = 0 2/t(1-t) gives 2 in the formula for S(n), and the coefficient at t^n in 2(1-t)^(1/2)/(t(1-t)) = 2/t*(1-t)^(-1/2) gives the second term. Alec Mihailovs http://webpages.shepherd.edu/amihailo/