Kepler-Bouwkamp Radius of Combinatorial Sequences

Tomislav Došlić Faculty of Civil Engineering University of Zagreb Kačićeva 26 10000 Zagreb Croatia

                        doslic@grad.hr

Abstract The Kepler-Bouwkamp constant is defined as the limit of radii of a sequence of concentric circles that are simultaneously inscribed in a regular n-gon and circumscribed around a regular (n + 1)-gon for n ≥ 3. The outermost circle, circumscribed around an equilateral triangle, has radius 1. We investigate what happens when the number of sides of regular polygons from the definition is given by a sequence different from the sequence of natural numbers.

1 Introduction

Take a unit circle and inscribe in it an equilateral triangle. Then inscribe a circle in that triangle, and inscribe an equilateral triangle in that circle. Continue the procedure. It is a high-school exercise to show that the sequence of radii of inscribed circles tends to zero as \frac{1}{2^{n}}. The conclusion remains valid if we replace triangle with any other regular polygon, only the rate of convergence is changed: For a regular m-gon, the radii tend to zero as (cos \frac{π}{m})^{n}. But what happens with the limiting radius if the number of sides of inscribed polygons is not constant? What if it increases so that the number of sides of n-th polygon is given as the n-th element of a sequence (a_{n}) of non-negative integers that are all greater than two? For a given regular n-gon, the ratio of radii of its inscribed and its circumscribed circle is equal to cos \frac{π}{n}. Hence the answer to our question will be given as an infinite product ∏_{a_{n}≥3}^{∞} cos \frac{π}{a_{n}}.

When a_{n} = n, the answer is well known: The limiting radius is equal to the Kepler-Bouwkamp constant ρ = ∏_{n=3}^{∞} cos \frac{π}{n} ≐ 0.1149420448 ([4, p. 428]; see also [7, \underline{A085365}]). To the best of my knowledge, there is only one other sequence, the sequence of odd primes, for which the answer was sought. Kitson [5] computed the limiting radius as ρ_{P} = 0.3128329295. The quantities in question were also computed for a few other sequences, mostly of the form n(n+1) for even and odd n, in a paper by Mathar [6], but they appear there as byproducts of some other computations. The goal of this note is to investigate for which sequences is that limit positive and to compute the limiting radii for several classes of combinatorial sequences. The results might be useful in further investigation of problems arising in computational geometry [6], and they might shed additional light on properties of integrals of the type ∫_{0}^{∞} ∏_{k=0}^{n} \frac{sin(a_{k}x)}{a_{k}x}dx [2]. They will also provide benchmarks for testing numerical methods for efficient evaluation of slowly convergent infinite products and series [3].

In the rest of this paper we consider only (weakly) increasing sequences (a_{n}) of nonnegative integers. If a sequence has elements smaller than 3, we ignore them, along with the corresponding terms in any expressions and formulas. We call such sequences admissible. Whenever referring to a sequence that is not itself admissible, we mean its largest admissible subsequence.

Let (a_{n}) be an admissible sequence. Its Kepler-Bouwkamp radius is denoted by κ(a_{n}) and defined by

κ(a_{n}) = ∏_{n}^{∞} cos \frac{π}{a_{n}} .

Hence, κ(n) = ρ, κ(p_{n}) = ρ_{P} and κ(c) = 0 for any constant sequence a_{n} = c.

2 Growth rates and convergence

In this section we investigate under what conditions a combinatorial sequence has a positive Kepler-Bouwkamp radius. We start by some elementary observations.

Proposition 1. Let (a_{n}) be an admissible sequence such that lim_{n→∞} < ∞. Then κ(a_{n}) = 0.

Proof. If an admissible sequence is bounded, it must have a convergent subsequence. The condition of integrality means that this subsequence must be constant. Then the infinite product over that subsequence diverges toward zero and the claim follows.

Hence a sequence must grow without bounds to have a positive Kepler-Bouwkamp radius. Any sequence growing faster that the sequence of natural numbers grows fast enough:

Proposition 2. Let an admissible sequence (a_{n}) be a subsequence of ℕ. Then κ(a_{n}) > ρ > 0.

What about sequences growing slower than ℕ? The following example shows that linear growth, no matter how slow, still suffices for the positivity of κ(a_{n}).

Proposition 3. Let (a_{n}) = ⌊\frac{n}{k}⌋ for a fixed k ≥ 2, n ≥ 3k. Then κ(a_{n}) = ρ^{k}.

Proof. It is enough to look at the case k = 2. Since each term in the product is repeated twice, we have

κ (⌊\frac{n}{2}⌋) = ∏_{n=3}^{∞} (cos \frac{π}{n})^{2} = (∏_{n=3}^{∞} cos \frac{π}{n})^{2} = ρ^{2}.

By the same reasoning we can conclude that the infinite product will converge for all admissible sequences in which the number of repetitions of an element remains finite.

Proposition 4. Let (a_{n}) be an admissible sequence and let there be a k ∈ ℕ such that no element of ℕ appears in (a_{n}) more than k times. Then κ(a_{n}) > ρ^{k} > 0.

What happens when the number of repetitions of an element grows without bounds? We first consider a concrete example.

Proposition 5. κ(⌊\sqrt{n}⌋) = 0.

Proof. Let us look at the sequence ⌊\sqrt{n}⌋, i.e., to its admissible subsequence. It start with seven 3’s, continues with nine 4’s, then eleven 5’s and so on. In general, an integer m appears in it exactly 2m + 1 times. Hence,

κ(⌊\sqrt{n}⌋) = ∏_{m=3}^{∞} (cos \frac{π}{m})^{2m+1} = ρ (∏_{m=3}^{∞} (cos \frac{π}{m})^{m})^{2} .

Now, the above infinite product converges if and only if converges the series of its logarithms ∑_{m=3}^{∞} m ln cos \frac{π}{m}. By expanding ln cos \frac{π}{m} we obtain ln cos \frac{π}{m} ∼ −\frac{π^{2}}{2m^{2}} , and then m ln cos \frac{π}{m} ∼ −\frac{π^{2}}{2m}. Hence the series diverges, and the infinite product goes to zero.

From the above example we can conclude that the infinite product will converge even in cases when the number of repetitions grows without bounds, as long as the growth is sublinear. The sequence ⌊\sqrt{n}⌋ is a limiting case – if a sequence (a_{n}) grows faster than ⌊\sqrt{n}⌋, its Kepler-Bouwkamp radius will be positive. The following result summarizes our findings.

Theorem 6. κ(a_{n}) > 0 if and only if (a_{n}) grows faster than ⌊\sqrt{n}⌋.

3 Explicit formulas and (semi)numerical examples

In this section we consider some admissible sequences and compute their Kepler-Bouwkamp radii. We start with a class of sequences for which it is possible to give explicit formulas. To the best of my knowledge, the class is very narrow; it contains only integer multiples of the sequence of powers of two. Hence, all sequences of this class are of the type a_{n} = m · 2^{n} for some integer m. The result follows from a classical infinite product formula for sinc x.

The sinc function is defined by

sinc x = \begin{cases}\frac{sin x}{x}, & if x ≠ 0; \\ 1, & if x = 0; \end{cases} .

The following infinite product representation of sinc x was known already to Viète:

sinc x = ∏_{n=1}^{∞} cos \frac{x}{2^{n}} .

It immediately yields formulas for the Kepler-Bouwkamp radius of an integer multiple of the sequence of powers of two.

Theorem 7. κ(m · 2^{n}) = sinc \frac{π}{m}.

Here are the values for some small m.

Corollary 8.

κ(2 · 2^{n}) = \frac{2}{π};  κ(3 · 2^{n}) = \frac{3\sqrt{3}}{2π};  κ(4 · 2^{n}) = \frac{3\sqrt{2}}{π};
κ(5 · 2^{n}) = \frac{5\sqrt{5 − \sqrt{5}}}{2\sqrt{2}π};  κ(6 · 2^{n}) = \frac{3}{π} .

Let us now look at a general admissible sequence (a_{n}) with κ(a_{n}) > 0. We follow, with minor modifications, the approach outlined by Kitson [5]. First we take the logarithm of both sides in the expression for κ(a_{n}),

ln κ(a_{n}) = ∑_{n} ln cos \frac{π}{a_{n}} .

The summand ln cos \frac{π}{a_{n}} can be expanded into a series

ln cos \frac{π}{a_{n}} = − ∑_{k=1}^{∞} \frac{2^{2k}(2^{2k} − 1)}{2k(2k)!}|B_{2k}| (\frac{π}{a_{n}} )^{2k}

[1, Formula 4.3.72]. Here B_{k} denote the Bernoulli numbers. Their exponential generating function is given by

∑_{k=0}^{∞} \frac{B_{k}x^{k}}{k!} = \frac{x}{e^{x} − 1}.

We have the following representation of even Bernoulli numbers:

B_{2k} = (−1)^{k−1} \frac{2(2k)!}{(2π)^{2k}} ζ(2k)

[1, Formula 23.1.18]. It can be obtained by expanding Bernoulli polynomials B_{n}(x) into a cosine Fourier series

B_{2k}(x) = (−1)^{k−1} \frac{2(2k)!}{(2π)^{2k}} ∑_{l=1}^{∞} \frac{cos 2lπx}{l^{2k}}

and substituting x = 0. When this is plugged into expansion of ln cos \frac{π}{a_{n}} a lot of cancellation occurs and we end with

ln cos \frac{π}{a_{n}} = − ∑_{k=1}^{∞} \frac{2^{2k} − 1}{k}ζ(2k)\frac{1}{a_{n}^{2k}} .

Now we can exchange the order of summation in the double sum for ln κ(a_{n}). We obtain

ln κ(a_{n}) = − ∑_{k=1}^{∞} \frac{2^{2k} − 1}{k}ζ(2k) ∑_{n} \frac{1}{a_{n}^{2k}} .

Here the interior sum runs over all n such that a_{n} ≥ 3. Hence, we have proved the following result.

Theorem 9.

κ(a_{n}) = exp (− ∑_{k=1}^{∞} \frac{2^{2k} − 1}{k}ζ(2k) ∑_{a_{n}≥3} a_{n}^{−2k}) .

We see that all dependence on the sequence (a_{n}) is well isolated and contained in the sum ∑_{a_{n}≥3} a_{n}^{−2k}. For some sequences that sum can be expressed in closed formulas in terms of k. By plugging them into the formula from the above theorem, we obtain rapidly converging expressions for κ(a_{n}); hence the (semi)numerical in the section title.

For a_{n} = n the sum can be expressed in terms of zeta function, taking into account the corrections for terms smaller than 3: ∑_{a_{n}≥3} n^{−2k} = \frac{4^{k}(ζ(2k)−1)−1}{4^{k}}. Similarly, Kitson used the prime zeta function to express the analogous sum in [5]. Let us denote by Z(a_{n}) the sum ∑_{a_{n}≥3} a_{n}^{−2k} for the sequence (a_{n}). In the following proposition we present the values of Z(a_{n}) for several sequences that yield to this approach.

Proposition 10.

Z(2n) = 2^{−2k}(ζ(2k) − 1);
Z(2n + 1) = 2^{−2k}ζ(2k, \frac{3}{2});
Z(m · n + p) = m^{−2k}ζ(2k, \frac{m + p}{m})  for m ≥ 3;
Z(n^{m}) = ζ(2mk) − 1;
Z(m^{n}) = \frac{1}{m^{2k} − 1}  for m ≥ 3.

(Here ζ(2k, q) is the Hurwitz zeta function.)

The Kepler-Bouwkamp radii of several sequences now follow by plugging the above expressions into formula of Theorem 9 and evaluating the resulting sums.

Corollary 11.

κ(2n) = 0.4297802164;
κ(2n + 1) = 0.2674437781;
κ(n^{2}) = 0.6402929927;
κ(3^{n}) = 0.4662745790.

Here are the Kepler-Bouwkamp radii of some sequences as reported in Mathar’s paper [6].

Proposition 12.

κ(n(n + 1)) = 0.8154881209;
κ((2n + 1)(2n + 2)) = 0.8373758680;
κ(p_{n}p_{n+1}) = 0.9729664541  for odd primes.

We conclude the paper by reporting the numerical values of Kepler-Bouwkamp radii for several interesting combinatorial sequences.

Proposition 13.

κ(n!) = 0.8583138700;
κ((2n − 1)!!) = 0.4888521829;
κ((2n)!!) = 0.9218702724;
κ(n^{n}) = 0.7022723378.

The class of sequences that allow the seminumerical approach is probably much wider than reported here, but I am not aware of any simple way to decide if a given sequence belongs to it.

4 Acknowledgments

Partial support of the Ministry of Science, Education and Sport of the Republic of Croatia (Grants No. 037-0000000-2779 and 177-0000000-0884) is gratefully acknowledged. I am thankful to the referee for careful reading and correcting several mistakes.

References

[1] M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, 1972.

[2] D. Borwein and J. M. Borwein, Some remarkable properties of sinc and related integrals, Ramanujan J. 5 (2001) 73–89.

[3] M. Chamberland and A. Straub, On gamma quotients and infinite products, preprint,

   http://arxiv.org/abs/1309.3455, 2013.

[4] S. R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003.

[5] A. R. Kitson, The prime analog of the Kepler-Bouwkamp constant, Math. Gazette 92 (2008) 293–294.

[6] R. J. Mathar, Tightly circumscribed regular polygons, preprint,

   http://arxiv.org/abs/1301.6293, 2013.

[7] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, available electronically at

   http://oeis.org.

2000 Mathematics Subject Classification: Primary 11Z05; Secondary: 51M04, 51M25 Keywords: Kepler-Bouwkamp constant, infinite product.

(Concerned with sequences \underline{A000027}, \underline{A000040}, \underline{A000079}, \underline{A000142}, \underline{A000244}, \underline{A000290}, \underline{A000714}, \underline{A002378}, \underline{A005408}, \underline{A005843}, \underline{A007283}, and \underline{A085365}.)

Received July 30 2014; revised versions received November 4 2014; November 5 2014. Published in Journal of Integer Sequences, November 7 2014.

Return to Journal of Integer Sequences home page.