Leibniz series

The Leibniz series is a formula for approximating the circle number that Gottfried Wilhelm Leibniz developed in the years 1673–1676 and first published in 1682 in the journal Acta Eruditorum . It is: ${\displaystyle \pi }$

${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\dotsb ={\frac {\pi }{4}}}$.

This formula was already known to the Indian mathematician Madhava in the 14th century and to the Scottish mathematician Gregory before 1671, Leibniz rediscovered it for continental European mathematics.

The convergence of this infinite series follows directly from the Leibniz criterion . The convergence is logarithmic .

Convergence speed

The remainder of the sum after the summands is ${\displaystyle n}$

${\displaystyle R_{n}=\sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}-{\frac {\pi }{4}}=-\sum _{k=n}^{\infty }{\frac {(-1)^{k}}{2k+1}}}$.

With the error estimation of the Leibniz criterion, the following applies

${\displaystyle |R_{n}|\leq {\frac {1}{2n+1}}}$.

Closer examination even shows that

${\displaystyle |R_{n}|<{\frac {1}{4n}}}$.

With summands you can get decimal places with an error <0.5 in the -th decimal place: ${\displaystyle n}$ ${\displaystyle s}$ ${\displaystyle s}$

${\displaystyle s(n)=\lg(2n)}$.

The number of summands required for meaningful decimal places in the result is accordingly ${\displaystyle n}$ ${\displaystyle s}$

${\displaystyle n(s)={\frac {1}{2}}\cdot 10^{\textstyle s}}$.

A list of partial sums resulting from Leibniz's formula

With the help of the Leibniz series, an approximation of the circle number can be calculated because it is ${\displaystyle \pi }$

${\displaystyle \pi =4\cdot \sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=\lim \limits _{n\to \infty }\left(4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}\right)}$.

The following list shows the elements of the sequence of partial sums of the Leibniz series multiplied by 4.

Since the sequence only converges very slowly, it is not suitable for the efficient calculation of . ${\displaystyle \pi }$

n (number of calculated fractions)	${\displaystyle 4\cdot \sum _{k=0}^{n-1}{\frac {(-1)^{k}}{2k+1}}}$ (Result)	Ratio to number of circles (ppm)	${\displaystyle 4\cdot \sum _{k=0}^{n}{\frac {(-1)^{k}}{2k+1}}}$ (Result)	Ratio to number of circles (ppm)	Average	Ratio to number of circles (ppm)
2	2.6666666666666665	-151173.6368432249	3. 4666666666666668	+103474.2721038077	3. 0666666666666664	-23849.6823697086
4th	2.8952380952380952	-78417.0914297870	3. 3396825396825394	+63053.9690963421	3.1 174603174603175	-7681.5611667224
8th	3. 0170718170718169	-39636.2132995474	3. 2523659347188758	+35260.2305084034	3.1 347188758953464	-2187.9913955720
16	3. 0791533941974261	-19875.0335505845	3. 2003655154095472	+18707.9829565416	3.1 397594548034866	-583.5252970215
32	3.1 103502736986859	-9944.7583872491	3.1 718887352371476	+9643.5423009843	3.141 1195044679165	-150.6080431325
64	3.1 259686069732875	-4973.2885002301	3.1 569763589112720	+4896.7854899649	3.141 4724829422798	-38.2515051326
100	3.1 315929035585528	-3183.0192943105	3.1 514934010709905	+3151.5058038744	3.1415 431523147719	-15.7567452180
1000	3.14 05926538397928	-318.3098066064	3.14 25916543395429	+317.9918149504	3.141592 1540896679	-0.1589958280
10,000	3.141 4926535900429	-31.8309885389	3.141 6926435905430	+31.8278057582	3.1415926 485902927	-0.0015913904
100,000	3.1415 826535897935	-3.1830988617	3.141 6026534897941	+3.1830670312	3.1415926535 397936	-0.0000159154
1000000	3.14159 16535897930	-0.3183098862	3.14159 36535887932	+0.3183095679	3.141592653589 2931	-0.0000001592
10000000	3.141592 5535897928	-0.0318309887	3.141592 7535897827	+0.0318309853	3.1415926535897 878	-0.0000000017
100000000	3.1415926 435897932	-0.0031830988	3.1415926 635897931	+0.0031830988	3.141592653589793 1	+0.0000000000
1000000000	3.14159265 25897930	-0.0003183099	3.14159265 45897932	+0.0003183099	3.141592653589793 1	+0.0000000000

Convergence Acceleration

The Euler transformation series generated from the Leibniz series the faster convergent series ( Nicolas Fatio , 1705)

${\displaystyle {\frac {\pi }{4}}={\frac {1}{2}}\left(1+{\frac {1}{1\cdot 3}}+{\frac {1\cdot 2}{1\cdot 3\cdot 5}}+...+{\frac {1\cdot 2...n}{1\cdot 3\cdot 5...(2n+1)}}+...\right).}$

Slow convergence

Ultimately, the Leibniz series (even after reshaping) is only suitable to a limited extent for calculating the number of circles. Other series and methods that converge more rapidly are listed in the Circle Number article .

See also

• Wallisian product
• Arctangent

literature

• VI Bityutskov: Leibniz series . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ^ GW Leibniz, De vera proportione circuli ad quadratum circumscriptum in numeris rationalibus expressa, in: Acta Eruditorum , February 1682, 41–46; Gerhardt , Leibniz's mathematical writings V, Halle 1958, 118-122.