In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:
arctan
x
=
x
−
x
3
3
+
x
5
5
−
x
7
7
+
⋯
=
∑
k
=
0
∞
(
−
1
)
k
x
2
k
+
1
2
k
+
1
.
{\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{2k+1}}.}
This series converges in the complex disk
|
x
|
≤
1
,
{\displaystyle |x|\leq 1,}
except for
x
=
±
i
{\displaystyle x=\pm i}
(where
arctan
±
i
=
∞
{\displaystyle \arctan \pm i=\infty }
).
It was first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma (c. 1340 – c. 1425), the founder of the Kerala school, and is described in extant works by Nīlakaṇṭha Somayāji (c. 1500) and Jyeṣṭhadeva (c. 1530). Mādhava's work was unknown in Europe, and the arctangent series was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673. In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series).
The special case of the arctangent of
1
{\displaystyle 1}
is traditionally called the Leibniz formula for π, or recently sometimes the Mādhava–Leibniz formula:
π
4
=
arctan
1
=
1
−
1
3
+
1
5
−
1
7
+
⋯
.
{\displaystyle {\frac {\pi }{4}}=\arctan 1=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots .}
The extremely slow convergence of the arctangent series for
|
x
|
≈
1
{\displaystyle |x|\approx 1}
makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed
1
4
π
{\displaystyle {\tfrac {1}{4}}\pi }
as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for
π
{\displaystyle \pi }
. Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.
Contents
Proof
If
y
=
arctan
x
{\displaystyle y=\arctan x}
then
tan
y
=
x
.
{\displaystyle \tan y=x.}
The derivative is
d
x
d
y
=
sec
2
y
=
1
+
tan
2
y
.
{\displaystyle {\frac {dx}{dy}}=\sec ^{2}y=1+\tan ^{2}y.}
Taking the reciprocal,
Convergence
The series for
arctan
′
{\textstyle \arctan '}
and
arctan
{\displaystyle \arctan }
converge within the complex disk
|
x
|
<
1
{\displaystyle |x|<1}
, where both functions are holomorphic. They diverge for
|
x
|
>
1
{\displaystyle |x|>1}
because when
x
=
±
i
{\displaystyle x=\pm i}
, there is a pole:
1
1
+
i
2
Accelerated series
Isaac Newton accelerated the convergence of the arctangent series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler's 1755 textbook; Euler wrote two proofs in 1779), yielding a series converging for
|
x
|
<
∞
,
{\textstyle |x|<\infty ,}
arctan
x
=
x
1
+
x
2
∑
n
=
0
∞
∏
k
=
1
n
2
k
2
k
+
1
History
The earliest person to whom the series can be attributed with confidence is Mādhava of Sangamagrāma (c. 1340 – c. 1425). The original reference (as with much of Mādhava's work) is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for
arctan
{\displaystyle \arctan }
include Nīlakaṇṭha Somayāji's Tantrasaṅgraha (c. 1500),
Jyeṣṭhadeva's Yuktibhāṣā (c. 1530), and the Yukti-dipika commentary by Sankara Variyar, where it is given in verses 2.206 – 2.209.