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---
title: "Approximations of pi"
chunk: 8/10
source: "https://en.wikipedia.org/wiki/Approximations_of_pi"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T16:19:48.727542+00:00"
instance: "kb-cron"
---
=== Continued fractions ===
Besides its simple continued fraction representation [3; 7, 15, 1, 292, 1, 1, ...], which displays no discernible pattern, π has many generalized continued fraction representations generated by a simple rule, including these two.
π
=
3
+
1
2
6
+
3
2
6
+
5
2
6
+
⋱
{\displaystyle \pi ={3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+\ddots \,}}}}}}}}
π
=
4
1
+
1
2
3
+
2
2
5
+
3
2
7
+
4
2
9
+
⋱
=
3
+
1
2
5
+
4
2
7
+
3
2
9
+
6
2
11
+
5
2
13
+
⋱
{\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}=3+{\cfrac {1^{2}}{5+{\cfrac {4^{2}}{7+{\cfrac {3^{2}}{9+{\cfrac {6^{2}}{11+{\cfrac {5^{2}}{13+\ddots }}}}}}}}}}}
The remainder of the Madhava–Leibniz series can be expressed as generalized continued fraction as follows.
π
=
4
∑
n
=
1
m
(
−
1
)
n
−
1
2
n
−
1
+
2
(
−
1
)
m
2
m
+
1
2
2
m
+
2
2
2
m
+
3
2
2
m
+
⋱
(
m
=
1
,
2
,
3
,
…
)
{\displaystyle \pi =4\sum _{n=1}^{m}{\frac {(-1)^{n-1}}{2n-1}}+{\cfrac {2(-1)^{m}}{2m+{\cfrac {1^{2}}{2m+{\cfrac {2^{2}}{2m+{\cfrac {3^{2}}{2m+\ddots }}}}}}}}\qquad (m=1,2,3,\ldots )}
Note that Madhava's correction term is
2
2
m
+
1
2
2
m
+
2
2
2
m
=
4
m
2
+
1
4
m
3
+
5
m
{\displaystyle {\frac {2}{2m+{\frac {1^{2}}{2m+{\frac {2^{2}}{2m}}}}}}=4{\frac {m^{2}+1}{4m^{3}+5m}}}
.
The well-known values ⁠22/7⁠ and ⁠355/113⁠ are respectively the second and fourth continued fraction approximations to π.
=== Trigonometry ===
==== Gregory–Leibniz series ====
The Gregory–Leibniz series
π
=
4
∑
n
=
0
∞
(
−
1
)
n
2
n
+
1
=
4
(
1
1
−
1
3
+
1
5
−
1
7
+
−
⋯
)
{\displaystyle \pi =4\sum _{n=0}^{\infty }{\cfrac {(-1)^{n}}{2n+1}}=4\left({\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+-\cdots \right)}
is the power series for arctan(x) specialized to x = 1. It converges too slowly to be of practical interest. However, the power series converges much faster for smaller values of
x
{\displaystyle x}
, which leads to formulae where
π
{\displaystyle \pi }
arises as the sum of small angles with rational tangents, known as Machin-like formulae.
==== Arctangent ====
Knowing that 4 arctan 1 = π, the formula can be simplified to get:
π
=
2
(
1
+
1
3
+
1
⋅
2
3
⋅
5
+
1
⋅
2
⋅
3
3
⋅
5
⋅
7
+
1
⋅
2
⋅
3
⋅
4
3
⋅
5
⋅
7
⋅
9
+
1
⋅
2
⋅
3
⋅
4
⋅
5
3
⋅
5
⋅
7
⋅
9
⋅
11
+
⋯
)
=
2
∑
n
=
0
∞
n
!
(
2
n
+
1
)
!
!
=
∑
n
=
0
∞
2
n
+
1
n
!
2
(
2
n
+
1
)
!
=
∑
n
=
0
∞
2
n
+
1
(
2
n
n
)
(
2
n
+
1
)
=
2
+
2
3
+
4
15
+
4
35
+
16
315
+
16
693
+
32
3003
+
32
6435
+
256
109395
+
256
230945
+
⋯
{\displaystyle {\begin{aligned}\pi &=2\left(1+{\cfrac {1}{3}}+{\cfrac {1\cdot 2}{3\cdot 5}}+{\cfrac {1\cdot 2\cdot 3}{3\cdot 5\cdot 7}}+{\cfrac {1\cdot 2\cdot 3\cdot 4}{3\cdot 5\cdot 7\cdot 9}}+{\cfrac {1\cdot 2\cdot 3\cdot 4\cdot 5}{3\cdot 5\cdot 7\cdot 9\cdot 11}}+\cdots \right)\\&=2\sum _{n=0}^{\infty }{\cfrac {n!}{(2n+1)!!}}=\sum _{n=0}^{\infty }{\cfrac {2^{n+1}n!^{2}}{(2n+1)!}}=\sum _{n=0}^{\infty }{\cfrac {2^{n+1}}{{\binom {2n}{n}}(2n+1)}}\\&=2+{\frac {2}{3}}+{\frac {4}{15}}+{\frac {4}{35}}+{\frac {16}{315}}+{\frac {16}{693}}+{\frac {32}{3003}}+{\frac {32}{6435}}+{\frac {256}{109395}}+{\frac {256}{230945}}+\cdots \end{aligned}}}
with a convergence such that each additional 10 terms yields at least three more digits.
π
=
2
+
1
3
(
2
+
2
5
(
2
+
3
7
(
2
+
⋯
)
)
)
{\displaystyle \pi =2+{\frac {1}{3}}\left(2+{\frac {2}{5}}\left(2+{\frac {3}{7}}\left(2+\cdots \right)\right)\right)}
This series is the basis for a decimal spigot algorithm by Rabinowitz and Wagon.
Another formula for
π
{\displaystyle \pi }
involving arctangent function is given by
π
2
k
+
1
=
arctan
⁡
2
−
a
k
−
1
a
k
,
k
≥
2
,
{\displaystyle {\frac {\pi }{2^{k+1}}}=\arctan {\frac {\sqrt {2-a_{k-1}}}{a_{k}}},\qquad \qquad k\geq 2,}
where
a
k
=
2
+
a
k
−
1
{\displaystyle a_{k}={\sqrt {2+a_{k-1}}}}
such that
a
1
=
2
{\displaystyle a_{1}={\sqrt {2}}}
. Approximations can be made by using, for example, the rapidly convergent Euler formula
arctan
⁡
(
x
)
=
∑
n
=
0
∞
2
2
n
(
n
!
)
2
(
2
n
+
1
)
!
x
2
n
+
1
(
1
+
x
2
)
n
+
1
.
{\displaystyle \arctan(x)=\sum _{n=0}^{\infty }{\frac {2^{2n}(n!)^{2}}{(2n+1)!}}\;{\frac {x^{2n+1}}{(1+x^{2})^{n+1}}}.}
Alternatively, the following simple expansion series of the arctangent function can be used
arctan
⁡
(
x
)
=
2
∑
n
=
1
∞
1
2
n
−
1
a
n
(
x
)
a
n
2
(
x
)
+
b
n
2
(
x
)
,
{\displaystyle \arctan(x)=2\sum _{n=1}^{\infty }{{\frac {1}{2n-1}}{\frac {{{a}_{n}}\left(x\right)}{a_{n}^{2}\left(x\right)+b_{n}^{2}\left(x\right)}}},}
where
a
1
(
x
)
=
2
/
x
,
b
1
(
x
)
=
1
,
a
n
(
x
)
=
a
n
−
1
(
x
)
(
1
−
4
/
x
2
)
+
4
b
n
−
1
(
x
)
/
x
,
b
n
(
x
)
=
b
n
−
1
(
x
)
(
1
−
4
/
x
2
)
−
4
a
n
−
1
(
x
)
/
x
,
{\displaystyle {\begin{aligned}&a_{1}(x)=2/x,\\&b_{1}(x)=1,\\&a_{n}(x)=a_{n-1}(x)\,\left(1-4/x^{2}\right)+4b_{n-1}(x)/x,\\&b_{n}(x)=b_{n-1}(x)\,\left(1-4/x^{2}\right)-4a_{n-1}(x)/x,\end{aligned}}}
to approximate
π
{\displaystyle \pi }
with even more rapid convergence. Convergence in this arctangent formula for
π
{\displaystyle \pi }
improves as integer
k
{\displaystyle k}
increases.
The constant
π
{\displaystyle \pi }
can also be expressed by infinite sum of arctangent functions as
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