A set of the Vi\`ete-like recurrence relations for the unity constant

TL;DR

This paper introduces a set of Viète-like recurrence relations for the constant 1, derived from nested radicals related to pi, demonstrating rapid convergence to unity through computational tests.

Contribution

It presents a novel set of recurrence relations for the constant 1 based on a Viète-like formula involving nested radicals, expanding the mathematical understanding of such relations.

Findings

Recurrence relations converge rapidly to 1

Derived from nested radicals related to pi

Validated through computational tests

Abstract

Using a simple Vi\`ete-like formula for $\pi$ based on the nested radicals $a_k = \sqrt{2 + a_{k-1}}$ and $a_1 = \sqrt{2}$, we derive a set of the recurrence relations for the constant $1$. Computational test shows that application of this set of the Vi\`ete-like recurrence relations results in a rapid convergence to unity.

Tables

Table 1. Table 1. The variable c k subscript 𝑐 𝑘 c_{k} and error term ϵ k subscript italic-ϵ 𝑘 \epsilon_{k} at index k 𝑘 k ranging from 4 4 4 to 15 15 15 .

$k$	$c_k$	${ϵ}_k$
4	0.90634716901914715794…	0.09365283098085284205…
5	0.95207914670092534858…	0.04792085329907465141…
6	0.97575264993237653232…	0.02424735006762346767…
7	0.98780284145152917070…	0.01219715854847082929…
8	0.99388282491415211156…	0.00611717508584788843…
9	0.99693673501114949604…	0.00306326498885050395…
10	0.99846719455859369106…	0.00153280544140630893…
11	0.99923330359286120490…	0.00076669640713879509…
12	0.99961657831851611515…	0.00038342168148388484…
13	0.99980827078273533526…	0.00019172921726466473…
14	0.99990413079635610519…	0.00009586920364389480…
15	0.99995206424931502866…	0.00004793575068497133…

Equations

$$\frac{2}{\pi }=\frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots .$$

$$a_1=\sqrt{2},$$

$$a_2=\sqrt{2+\sqrt{2}},$$

$$a_3=\sqrt{2+\sqrt{2+\sqrt{2}}}$$

$$⋮$$

$$a_k=\underset{k\text{square}\text{roots}}{\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}}$$

$$\frac{2}{\pi }=\underset{k\to \infty }{\lim }\prod _{k=1}^K\frac{a_k}{2}.$$

$$\frac{\pi }{2^{k+1}}=\arctan \left(\frac{\sqrt{2-a_{k-1}}}{a_k}\right),k\ge 2,$$

$$\frac{\pi }{2^3}+\frac{\pi }{2^4}+\frac{\pi }{2^5}\cdots$$

$$\arctan \left(\frac{\sqrt{2-a_1}}{a_2}\right)+\arctan \left(\frac{\sqrt{2-a_2}}{a_3}\right)+\arctan \left(\frac{\sqrt{2-a_3}}{a_4}\right)+\cdots$$

$$\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}\cdots =\frac{1}{4}$$

$$\frac{\pi }{4}=\underset{K\to \infty }{\lim }\sum _{k=1}^K\arctan \left(\frac{\sqrt{2-a_k}}{a_{k+1}}\right).$$

$$\frac{\pi }{4}$$

$$+\arctan \left(\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}\right)+\cdots$$

$$\frac{\pi }{4}$$

$$=\underset{K\to \infty }{\lim }\sum _{k=1}^K\arctan \left(b_k\right),$$

$$b_k=\frac{\sqrt{2-a_k}}{a_{k+1}}$$

$$a_k=\sqrt{2+a_{k-1}},a_1=\sqrt{2}.$$

$$\arctan \left(1\right)=\frac{\pi }{4}$$

$$\arctan \left(1\right)=\underset{K\to \infty }{\lim }\sum _{k=1}^K\arctan \left(b_k\right).$$

$$\arctan \left(x\right)+\arctan \left(y\right)=\arctan \left(\frac{x+y}{1-xy}\right)$$

$$c_k=\frac{c_{k-1}+b_k}{1-c_{k-1}{b}_k},c_1=b_1.$$

$$\arctan \left(1\right)=\arctan \left(c_k\right)+\underset{L\to \infty }{\lim }\sum _{\ell =k+1}^L\arctan \left(b_{\ell }\right).$$

$$\arctan \left(b_{\ell }\right)=b_{\ell }-\frac{b_{\ell }^3}{3}+\frac{b_{\ell }^5}{5}-\frac{b_{\ell }^7}{7}+\cdots =b_{\ell }+O\left(b_{\ell }^3\right).$$

$$\underset{\ell \to \infty }{\lim }\frac{b_{\ell +1}}{b_{\ell }}=\underset{\ell \to \infty }{\lim }\frac{\arctan \left(b_{\ell +1}\right)}{\arctan \left(b_{\ell }\right)}=\underset{\ell \to \infty }{\lim }\frac{\pi /2^{\ell +2}}{\pi /2^{\ell +1}}=\frac{1}{2}.$$

$$\left\{b_1,b_2,b_3,\dots ,b_{\ell },\dots \right\}.$$

$$\frac{b_2}{b_1}<\frac{b_3}{b_2}<\frac{b_4}{b_3}<\cdots <\frac{b_{\ell +1}}{b_{\ell }}<\cdots <\frac{1}{2}.$$

$$\arctan \left(1\right)=\underset{k\to \infty }{\lim }\left[\arctan \left(c_k\right)+\underset{L\to \infty }{\lim }\sum _{\ell =k+1}^L{b}_{\ell }\right].$$

$$\left\{b_1,\frac{b_1}{2},\frac{b_1}{2^2},\frac{b_1}{2^3},\cdots \frac{b_1}{2^{\ell }}\cdots \right\}$$

$$\sum _{\ell =k+1}^L{b}_{\ell }<\sum _{\ell =k+1}^L\frac{b_1}{2^{\ell -1}},L>k>0,$$

$$\underset{L\to \infty }{\lim }\sum _{\ell =k+1}^L\frac{b_1}{2^{\ell -1}}\to 0,k\to \infty ,$$

$$\underset{L\to \infty }{\lim }\sum _{\ell =k+1}^L{b}_{\ell }\to 0,k\to \infty .$$

Full text

Keywords: arctangent function, constant pi, constant 1

1 Description and implementation

1.1 Derivation

Several centuries ago the French mathematician François Viète derived a remarkable formula for pi

[TABLE]

Nowadays this well-known equation is commonly regarded as the Viète’s formula for pi [1, 2, 3, 4]. The uniqueness of this formula is due to nested radicals consisting of square roots of twos only. Defining these nested radicals as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

the Viète’s formula (1) for pi can be rewritten in a compact form as follows

[TABLE]

There is a simple Viète-like formula for pi that can be represented in form [5]

[TABLE]

From this formula it follows that

[TABLE]

and because of the decreasing geometric series

[TABLE]

the equation (3) can be expressed in a more simplified form

[TABLE]

It is more convenient for our purpose to represent the equation (4) as

[TABLE]

or

[TABLE]

where the arguments of the arctangent functions can be found by using the recurrence relations

[TABLE]

and

[TABLE]

Since

[TABLE]

we can also write

[TABLE]

The right side of the equation (5) consists of the infinite summation terms of the arctangent functions. We may attempt to exclude the infinite sum using the identity

[TABLE]

repeatedly. Specifically, we employ the following recurrence relations that just reflects the successive application of the identity (6) above

[TABLE]

This enables us to rewrite the equation (5) as

[TABLE]

According to the Maclaurin expansion series

[TABLE]

Since at $\ell\to\infty$ the variable ${{b}_{\ell}}\to 0$ and, therefore, due to negligible $O\left(b_{\ell}^{3}\right)$ we can simply replace it by $\arctan\left({{b}_{\ell}}\right)$ and then use the equation (2) in order to find a ratio of the limit

[TABLE]

Consider the following infinite sequence

[TABLE]

According to the limit (8) the ratio ${{b}_{\ell+1}}/{{b}_{\ell}}$ tends to $1/2$ with increasing index $\ell$. Consequently, it is not difficult to see now that

[TABLE]

In fact, the tendency of the ratio ${{b}_{\ell+1}}/{{b}_{\ell}}$ towards $1/2$ with increasing index $\ell$ is very fast. In particular, when the index $\ell$ is large enough, say at $\ell>10$, the sequence (9) behaves almost like a decreasing geometric progression where a common ratio is $1/2$.

Since the index $k$ in the equation (7) can be taken arbitrarily large, we can rewrite it in form

[TABLE]

Taking into account that the ratio ${{b}_{\ell+1}}/{{b}_{\ell}}$ tends to but never exceeds $1/2$, we can conclude that the damping rate in the sequence (9) is faster than that of in a decreasing geometric progression

[TABLE]

with fixed common ratio $1/2$. This signifies that

[TABLE]

and since the limit of the decreasing geometric series

[TABLE]

we prove that

[TABLE]

As a consequence, the equation (10) can be further simplified as

[TABLE]

Thus, we can infer that the constant $1$ can be approached successively by increment of the index $k$ in a set of the Viète-like recurrence relations

[TABLE]

such that ${{c}_{k\to\infty}}\to 1.$

1.2 Computation

Consider the first three elements from the sequence (9)

[TABLE]

[TABLE]

and

[TABLE]

Consequently, the corresponding first three values of the variable ${{c}_{k}}$ are

[TABLE]

[TABLE]

and

[TABLE]

respectively.

From these examples one can see that the set (11) of the Viète-like recurrence relations gradually builds the continued fractions in the numerator and denominator of the variable ${{c}_{k}}$ at each successive step in increment of the index $k$. It is also interesting to note that each value of the variable ${{c}_{k}}$ is based on nested radicals consisting of square roots of twos only.

Figure 1 shows the dependence of the variables ${{a}_{k}}$, ${{b}_{k}}$ and ${{c}_{k}}$ as a function of the index $k$ by blue, green and red colors, respectively. We can observe how the variable ${{c}_{k}}$ tends to $1$ while the variables ${{a}_{k}}$ and ${{b}_{k}}$ tend to $2$ and [math], respectively.

Table 1 shows the values of variable ${{c}_{k}}$ and error term ${{\varepsilon}_{k}}=1-{{c}_{k}}$ with corresponding index $k$ ranging from $4$ to $15.$ As we can see from this table, the variable ${{c}_{k}}$ quite rapidly tends to unity with increasing index $k$. In particular, the error term ${{\varepsilon}_{k}}$ decreases by factor of about $2$ at each increment of the index $k$ by one.

2 New formula for pi

As the error term ${{\varepsilon}_{k}}$ decreases successively by factor of about $2$ (see third column in the Table 1), we may expect that $2^{k}\varepsilon_{k}$ is convergent and tends to some constant when the index $k$ tends to infinity. The computational test shows that the value $2^{k}\varepsilon_{k}$ approaches to $\pi/2$ as the index $k$ increases. Therefore, we assume that

[TABLE]

or

[TABLE]

Furthermore, relying on numerical results we also suggest a generalization to the power $m$ as given by

[TABLE]

Since the variable $c_{k}$ is determined within the set (11) of the Viète-like recurrence relations, the new equation (12) can also be regarded as the Viète-like formula for pi.

3 Conclusion

We show a set (11) of the Viète-like recurrence relations for the constant $1$ derived by using the Viète-like formula (2) for pi. Sample computations reveal that the variable ${{c}_{k}}$ quite rapidly tends to unity as the index $k$ increases.

Acknowledgments

This work is supported by National Research Council Canada, Thoth Technology Inc. and York University.