Leibniz Formula For π
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Leibniz formula for , named after
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
, states that \frac = 1-\frac+\frac-\frac+\frac-\cdots = \sum_^ \frac, an alternating series. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was later independently rediscovered by James Gregory in 1671 and Leibniz in 1673. The Taylor series for the inverse tangent function, often called '' Gregory's series'', is \arctan x = x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac. The Leibniz formula is the special case \arctan 1 = \tfrac14\pi. It also is the Dirichlet -series of the non-principal Dirichlet character of modulus 4 evaluated at s=1, and therefore the value of the Dirichlet beta function.


Proofs


Proof 1

\begin \frac &= \arctan(1) \\ &= \int_0^1 \frac 1 \, dx \\ pt& = \int_0^1\left(\sum_^n (-1)^k x^+\frac\right) \, dx \\ pt& = \left(\sum_^n \frac\right) +(-1)^ \left(\int_0^1\frac \, dx\right) \end Considering only the integral in the last term, we have: 0 \le \int_0^1 \frac\,dx \le \int_0^1 x^\,dx = \frac \;\rightarrow 0 \text n \rightarrow \infty. Therefore, by the squeeze theorem, as , we are left with the Leibniz series: \frac4 = \sum_^\infty\frac


Proof 2

Let f(z) = \sum_^\fracz^, when , z, <1, the series \sum_^\infty (-1)^k z^ converges uniformly, then \arctan(z) = \int_^ \frac dt =\sum_^\fracz^ = f(z) \ (, z, <1). Therefore, if f(z) approaches f(1) so that it is continuous and converges uniformly, the proof is complete, where, the series \sum_^\frac to be converges by the Leibniz's test, and also, f(z) approaches f(1) from within the Stolz angle, so from Abel's theorem this is correct.


Convergence

Leibniz's formula converges extremely slowly: it exhibits sublinear convergence. Calculating to 10 correct decimal places using direct summation of the series requires precisely five billion terms because for (one needs to apply Calabrese error bound). To get 4 correct decimal places (error of 0.00005) one needs 5000 terms. Even better than Calabrese or Johnsonbaugh error bounds are available. However, the Leibniz formula can be used to calculate to high precision (hundreds of digits or more) using various convergence acceleration techniques. For example, the Shanks transformation, Euler transform or Van Wijngaarden transformation, which are general methods for alternating series, can be applied effectively to the partial sums of the Leibniz series. Further, combining terms pairwise gives the non-alternating series \frac = \sum_^ \left(\frac-\frac\right) = \sum_^ \frac which can be evaluated to high precision from a small number of terms using Richardson extrapolation or the Euler–Maclaurin formula. This series can also be transformed into an integral by means of the Abel–Plana formula and evaluated using techniques for
numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
.


Unusual behaviour

If the series is truncated at the right time, the
decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\cdots b_0.a_1a_2\cdots Here is the decimal separator ...
of the approximation will agree with that of for many more digits, except for isolated digits or digit groups. For example, taking five million terms yields 3.141592\underline5358979323846\underline643383279502\underline841971693993\underline058... where the underlined digits are wrong. The errors can in fact be predicted; they are generated by the Euler numbers according to the
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates Limit of a function#Limits at infinity, tends to infinity. In pro ...
formula \frac - 2 \sum_^ \frac \sim \sum_^\infty \frac where is an integer divisible by 4. If is chosen to be a power of ten, each term in the right sum becomes a finite decimal fraction. The formula is a special case of the Euler–Boole summation formula for alternating series, providing yet another example of a convergence acceleration technique that can be applied to the Leibniz series. In 1992, Jonathan Borwein and Mark Limber used the first thousand Euler numbers to calculate to 5,263 decimal places with the Leibniz formula.


Euler product

The Leibniz formula can be interpreted as a
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...
using the unique non-principal Dirichlet character modulo 4. As with other Dirichlet series, this allows the infinite sum to be converted to an infinite product with one term for each
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. Such a product is called an Euler product. It is: \begin \frac\pi4 &= \biggl(\prod_\frac\biggr) \biggl( \prod_\frac\biggr) \\ mu &= \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdot \frac \cdots \end In this product, each term is a superparticular ratio, each numerator is an odd prime number, and each denominator is the nearest multiple of 4 to the numerator.. The product is conditionally convergent; its terms must be taken in order of increasing .


See also

* List of formulae involving


References

{{DEFAULTSORT:Leibniz formula for Pi Pi algorithms Articles containing proofs Gottfried Wilhelm Leibniz Series (mathematics)