A formula for pi involving nested radicals

Transcription

1 arxiv: v [math.gm] 7 Apr 08 A formula for pi involving nested radicals S. M. Abrarov and B. M. Quine April 7, 08 Abstract We present a new formula for pi involving nested radicals with rapid convergence. This formula is based on the arctangent function identity with argument x = a k /a k, where a k = } {{ } k square roots is a nested radical consisting of k square roots. The computational test we performed reveals that the proposed formula for pi provides a significant improvement in accuracy as the integer k increases. Keywords: constant pi, arctangent function, nested radical MSC classes: Y60 Introduction In 593 the French mathematician François Viéte discovered a classical formula for the constant pi that can be expressed elegantly in nested radicals consisting of square roots of twos [,, 3] = Dept. Earth and Space Science and Engineering, York University, Toronto, Canada, M3J P3. Dept. Physics and Astronomy, York University, Toronto, Canada, M3J P3.

2 He found this formula for pi geometrically by considering a regular polygon enclosed inside the circle with unit radius. It is convenient to define the nested radicals by recurrence relations a k = +a k and a = to represent the Viéte s formula for pi in a more compact form as = lim K Many interesting formulas involving nested radicals consisting of square roots of twos have been reported in the modern literature see for example [,, 3]. In this paper we derive a new formula for pi that is also based on this type of the nested radicals = k+ i m= m K k= a k. +i a k / a k m i a k / a k m The computational test reveals that accuracy of the constant pi in the truncated formula can be considerably improved by successive increment of the integer k. Derivation As it has been shown in our recent paper [4], any function f t differentiable within the interval t [0,] can be integrated numerically by truncating the parameters L or M in the following limits see equations 9 and 6 in [4]. 0 f tdt = lim L M l= m=0 m + L m+ m+! fm t l / t= L 3a and 0 f tdt = lim M M l= m=0 m + L m+ m+! fm t t= l / L, 3b

3 respectively. Since substituting the integrand 0 x +x tdt = arctanx f t = x +x t into the equation 3a results in see [5] for more details in derivation arctanx = i lim L l= M + m= m l +il/x m l il/x m Accordingtoequations3aand3btheparametersLandM underthelimit notation are interchangeable compare equations 9 and 6 from the paper [4]. Consequently, using absolutely same derivation procedure as described in [5] for the equation 4a above, we can also write arctanx = i lim M l= M + m= m l +il/x m l il/x m Comparing equations 4a and 4b we can see that at least one of the parameters L or M must be large enough in truncation for high-accuracy approximation. However, as the parameter M is more important for rapid convergence, the limit 4b is preferable for numerical analysis. Since in the limit 4b the integer L may not be necessarily large, in order to simplify it we can choose any small value, say L =. This leads to the equation M + arctanx = i lim M m= m +i/x m i/x m Since the integer M tends to infinity, the upper bound M/ + of summation also tends to infinity. Consequently, this equation can be rewritten in a simplified form arctanx = i m= m +i/x m i/x m a 4b. 5

4 Using the identity for the cosine double-angle cos = cos k k+ and taking into consideration that cos =, we can readily find by induction + cos =, cos =, cos =, 5 where. cos = k+ }{{} k square roots Consider the following relation = arctan tan = arctan k+ k+ sin k+ cos k+, 7 sin = cos. 8 k+ k+ Substituting the equation 6 into the identity 8 leads to sin = k }{{} k square roots 4

5 Consequently, from the equations 6, 7 and 9 we obtain a simple formula for the constant pi }{{} k square roots = arctan k }{{} k square roots or = arctan k+ ak a k. 0 Lastly, combining equations 5 and 0 together results in the formula for pi. 3 Algorithmic implementation 3. Methodology description As it has been reported previously in the paper [5], the decrease of the argument x in the limit 4a improves significantly the accuracy in computing pi. Therefore, we may also expect a considerable improvement in accuracy of the arctangent function identity 5 when its argument x decreases. In fact, the equation is based on the arctangent function identity 5 when its argument x is equal to a k /a k. The increment of the integer k by one decreases the argument x = a k /a k by a factor that tends to two as k. Therefore, the value of argument x = a k /a k decreases very rapidly in a geometric progression as the integer k increases. As a consequence, this approach leads to a significant improvement in accuracy of the constant pi. 5

6 3. Computational results In order to estimate the convergence rate, we performed sample computations of the constant pi by using a rearranged form of the equation as follows = m max k+ i +ε, m= m +i a k / a k m i a k / a k m where m max >> is the truncating integer and ε is the error term. The computational test reveals that even at smallest values of the integer k the equation can be quite rapid in convergence. In particular, at m max = 5 and k equals only to, and 3 we can observe a relatively large overlap in digits coinciding with actual value of the constant pi as given by for k = we imply that a 0 = 0 and }{{ } coinciding digits }{{ } coinciding digits }{{ } , 05 coinciding digits respectively. Further, in algorithmic implementation we incremented k by one at each successive step while keeping value of the parameter m max fixed and equal to 69. Thus, the computational test we performed shows that at k equal to 5, 6, 7, 8, 9 and 0, the error term ε by absolute value becomes equal to , , , , and , respectively. As we can see, with only m max = 69 summation terms each increment of the integer k just by one contributes for more than 00 additional decimal digits of pi. Since the convergence improves while a k /a k decreases, it is not necessary to increment continuously the integer k. For example, when k is fixed and equal to 3, 50 and 90, each increment of the integer m max by one contributes for 4, 30 and 54 decimal digits of pi, respectively. 6

7 4 Theoretical analysis Proposition According to computational test that has been shown in the previous section, even if the parameter m max is fixed at 69, the accuracy of pi, nevertheless, improves continuously while the integer k increases. Therefore, relying on these experimental results we assume that the equation can be modified as m max = i lim k+ m m= +i a k / a k m i a K / a k m. Proof The proof is not difficult. Consider the following integral 0 k+ ak /a k a k /a k t dt = k+ arctan The integrand of this integral ak a k =. at the limit when k becomes g k t = k+ ak /a k a k /a k t lim g kt = g t =. Since the function g t is just a constant, only its zeroth order of the derivative g 0 t isnot equal to zero. This signifies that if thefunction g t is substituted into equation 3b, then it is no longer necessary to tend the integer M to infinity because for anyother thanzerothorder of thederivative we always get g m t = 0, m > 0. 7

8 Consequently, we can infer = lim g k tdt 0 = lim lim M = lim l= m=0 M l= m=0 M m + L m+ m+! g k m t l / t= L m + L m+ m+! g k m t t= l / L and from the equations 3 and 3b it follows now that 3 = i lim k+ l= M + m= m l +il a k / a k m l il a k / a k m. Implying now that m max = M/ +, at L = this limit is reduced to equation. This completes the proof. 4 Corollary At L = and M = the limit 4 is simplified as = i lim k+ +i a k / a k i a k / a k and since see equation 6 lim a k = lim cos = k+ the limit 5 provides = i lim k+ = lim +4i / a k 4i / a k 8/ ak k+ +6/ a k 8 = lim k+ ak +6/. a k 5 6 8

9 From the equation 8 it immediately follows that lim ak = lim sin k+ = 0. Consequently, the limit 6 can be simplified further as or = lim k+ 8 6/ a k = lim k a k = lim k This is a well-known formula for pi []. 5 Conclusion }{{} k square roots A new formula for pi based on the arctangent function identity 5 with argument x = a k /a k, where a k = , } {{ } k square roots is presented. This approach demonstrates high efficiency in computation due to rapid convergence. Specifically, the computational test reveals that with only 69 summation terms the increment of integer k just by one provides more than 00 additional decimal digits of the constant pi. Acknowledgments This work is supported by National Research Council Canada, Thoth Technology Inc. and York University. The authors thank the reviewers for constructive comments and recommendations. 9

10 References [] L.D. Servi, Nested square roots of, Amer. Math. Monthly, [] A. Levin, A new class of infinite products generalizing Viéte s product formula for, Ramanujan J [3] R. Kreminski, to thousands of digits from Vieta s formula, Math. Magazine, [4] S.M. Abrarov and B.M. Quine, Identities for the arctangent function by enhanced midpoint integration and the high-accuracy computation of pi, arxiv: , 06. [5] S.M. Abrarov and B.M. Quine, A simple identity for derivatives of the arctangent function, arxiv: , 06. 0