[math-fun] Help with a sum of a series
Hi all, I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are: 1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ... I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum? Thanks, Kerry
That sum is just c(1/4)/2, where c(x) = 2/(1 + sqrt(1 - 4x)) is the generating function for the Catalan numbers. So yes, it evaluates to 1. -- Adam P. Goucher
Sent: Tuesday, April 14, 2015 at 10:43 PM From: "Kerry Mitchell" <lkmitch@gmail.com> To: "Math Fun mailing list" <math-fun@mailman.xmission.com> Subject: [math-fun] Help with a sum of a series
Hi all,
I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are:
1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ...
I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The maple command sum(1/2*(2*n-2)!/(n-1)!/n!/(4^(n-1)),n=1..infinity); returns 1. On Tue, Apr 14, 2015 at 4:58 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
That sum is just c(1/4)/2, where c(x) = 2/(1 + sqrt(1 - 4x)) is the generating function for the Catalan numbers. So yes, it evaluates to 1.
-- Adam P. Goucher
Sent: Tuesday, April 14, 2015 at 10:43 PM From: "Kerry Mitchell" <lkmitch@gmail.com> To: "Math Fun mailing list" <math-fun@mailman.xmission.com> Subject: [math-fun] Help with a sum of a series
Hi all,
I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are:
1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ...
I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Nicer proof: We repeatedly flip a coin until there are strictly more tails than heads. The nth term in your series is the probability that this takes time exactly 2n + 1. Since the balanced random walk is a recurrent Markov process, the sum of these probabilities is precisely 1. Sincerely, Adam P. Goucher
Sent: Tuesday, April 14, 2015 at 10:58 PM From: "Adam P. Goucher" <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Help with a sum of a series
That sum is just c(1/4)/2, where c(x) = 2/(1 + sqrt(1 - 4x)) is the generating function for the Catalan numbers. So yes, it evaluates to 1.
-- Adam P. Goucher
Sent: Tuesday, April 14, 2015 at 10:43 PM From: "Kerry Mitchell" <lkmitch@gmail.com> To: "Math Fun mailing list" <math-fun@mailman.xmission.com> Subject: [math-fun] Help with a sum of a series
Hi all,
I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are:
1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ...
I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It comes from the generating function recurrence: c(x) = sum C_n x^n = 1 + xc(x)^2 = (1-sqrt(1-4x))/2x Plugging in x = 1/4, you get sum C_n/4^n = 2 On Tue, Apr 14, 2015 at 2:43 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Hi all,
I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are:
1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ...
I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Thank you all for your solutions. Very helpful! Kerry On Tue, Apr 14, 2015 at 3:08 PM, Mike Stay <metaweta@gmail.com> wrote:
It comes from the generating function recurrence:
c(x) = sum C_n x^n = 1 + xc(x)^2 = (1-sqrt(1-4x))/2x
Plugging in x = 1/4, you get sum C_n/4^n = 2
On Tue, Apr 14, 2015 at 2:43 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Hi all,
I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are:
1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ...
I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum?
Thanks, Kerry _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Adam P. Goucher -
James Buddenhagen -
Kerry Mitchell -
Mike Stay