Mathematical power series of arctangent
In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:
formula_1
This series converges in the complex disk formula_2 except for formula_3 (where 
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 &amp;NoBreak;&amp;NoBreak; is traditionally called the "Leibniz formula" for π, or recently sometimes the "Mādhava–Leibniz formula":
formula_4
The extremely slow convergence of the arctangent series for formula_5 makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed &amp;NoBreak;&amp;NoBreak; as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for &amp;NoBreak;&amp;NoBreak;. Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.
Proof.
If formula_6 then formula_7 The derivative is
formula_8
Taking the reciprocal,
formula_9
This sometimes is used as a definition of the arctangent:
formula_10
The Maclaurin series for formula_11 is a geometric series:
formula_12
One can find the Maclaurin series for formula_13 by naïvely integrating term-by-term:
formula_14
While this turns out correctly, integrals and infinite sums cannot always be exchanged in this manner. To prove that the integral on the left converges to the sum on the right for real formula_2 formula_16 can instead be written as the "finite" sum,
formula_17
Again integrating both sides, 
formula_18
In the limit as formula_19 the integral on the right above tends to zero when formula_2 because
formula_21
Therefore,
formula_22
Convergence.
The series for formula_23 and formula_13 converge within the complex disk formula_25, where both functions are holomorphic. They diverge for formula_26 because when formula_3, there is a pole:
formula_28
When formula_29 the partial sums formula_30 alternate between the values formula_31 and formula_32 never converging to the value formula_33
However, its term-by-term integral, the series for formula_34 (barely) converges when formula_35 because formula_16 disagrees with its series only at the point formula_37 so the difference in integrals can be made arbitrarily small by taking sufficiently many terms: 
formula_38
Because of its exceedingly slow convergence (it takes five billion terms to obtain 10 correct decimal digits), the Leibniz formula is not a very effective practical method for computing formula_39 Finding ways to get around this slow convergence has been a subject of great mathematical interest.
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
formula_40
where formula_41formula_42 and formula_43formula_44
Each term of this modified series is a rational function with its poles at formula_3 in the complex plane, the same place where the arctangent function has its poles. By contrast, a polynomial such as the Taylor series for arctangent forces all of its poles to infinity.
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 formula_13 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.
Notes.
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