In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:[1]
The special case of the arctangent of is traditionally called the Leibniz formula for π, or recently sometimes the Mādhava–Leibniz formula:
The extremely slow convergence of the arctangent series for makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for . Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.
Proof
If then The derivative is
Taking the reciprocal,
This sometimes is used as a definition of the arctangent:
One can find the Maclaurin series for by naïvely integrating term-by-term:
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 can instead be written as the finite sum,[4]
Again integrating both sides,
In the limit as the integral on the right above tends to zero when because
Therefore,
Convergence
The series for and converge within the complex disk , where both functions are holomorphic. They diverge for because when , there is a pole:
When the partial sums alternate between the values and never converging to the value
However, its term-by-term integral, the series for (barely) converges when because disagrees with its series only at the point so the difference in integrals can be made arbitrarily small by taking sufficiently many terms:
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 Finding ways to get around this slow convergence has been a subject of great mathematical interest.
Accelerated series
Isaac Newtonaccelerated the convergence of the arctangent series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler's 1755 textbook; Euler wrote two proofs in 1779), yielding a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 [5]
where and
Each term of this modified series is a rational function with its poles at 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.
K.V. Sarma (ed.). "Tantrasamgraha with English translation"(PDF) (in Sanskrit and English). Translated by V.S. Narasimhan. Indian National Academy of Science. p.48. Archived from the original(PDF) on 9 March 2012. Retrieved 17 January 2010.
Gupta, Radha Charan (1973). "The Mādhava–Gregory series". The Mathematics Education. 7: B67–B70.
Gupta, Radha Charan (1987). "South Indian Achievements in Medieval Mathematics". Ganịta Bhāratī. 9 (1–4): 15–40. Extension of a talk delivered at the Jodhpur University. Reprinted in Ramasubramanian, K., ed. (2019). Gaṇitānanda: Selected Works of Radha Charan Gupta on History of Mathematics. Springer. pp.417–442. doi:10.1007/978-981-13-1229-8_40.
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