In
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
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
The Leibniz formula is the special case
It also is the Dirichlet -series of the non-principal
Dirichlet character of modulus 4 evaluated at
and therefore the value of the
Dirichlet beta function.
Proofs
Proof 1
Considering only the integral in the last term, we have:
Therefore, by the
squeeze theorem, as , we are left with the Leibniz series:
Proof 2
Let
, when
, the series
converges uniformly, then
Therefore, if
approaches
so that it is continuous and converges uniformly, the proof is complete, where, the series
to be converges by the
Leibniz's test, and also,
approaches
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
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
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
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:
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)