Gregory's series

Short description: Mathematical power series defining arctan

In mathematics, Gregory's series for the inverse tangent function is its infinite Taylor series expansion at the origin:

[math]\displaystyle{ \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}. }[/math]

This series converges in the complex disk [math]\displaystyle{ |x| \leq 1, }[/math] except for [math]\displaystyle{ x = \pm i }[/math] (where [math]\displaystyle{ \arctan \pm i = \infty }[/math]).

It was first discovered in the 14th century by Madhava of Sangamagrama (c. 1340 – c. 1425), as credited by Madhava's Kerala school follower Jyeṣṭhadeva's Yuktibhāṣā (c. 1530). In recent literature it is sometimes called the Madhava–Gregory series to recognize Madhava's priority (see also Madhava series). It was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673, who obtained the Leibniz formula for π as the special case

[math]\displaystyle{ \frac\pi4 = \arctan 1 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots. }[/math]

Proof

The derivative of arctan x is 1 / (1 + x²); conversely, the integral of 1 / (1 + x²) is arctan x.

If [math]\displaystyle{ y = \arctan x }[/math] then [math]\displaystyle{ \tan y = x. }[/math] The derivative is

[math]\displaystyle{ \frac{dx}{dy} = \sec^2 y = 1 + \tan^2 y. }[/math]

Taking the reciprocal,

[math]\displaystyle{ \frac{dy}{dx} = \frac{1}{1 + \tan^2 y} = \frac{1}{1 + x^2}. }[/math]

This sometimes is used as a definition of the arctangent:

[math]\displaystyle{ \arctan x = \int_0^x\frac{du}{1 + u^2}. }[/math]

The Maclaurin series for [math]\displaystyle{ x \mapsto \arctan' x = 1 \big/ \left(1 + x^2\right) }[/math] is a geometric series:

[math]\displaystyle{ \frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots = \sum_{k=0}^\infty \bigl({-x^2}\bigr){\vphantom)}^k. }[/math]

One can find the Maclaurin series for [math]\displaystyle{ \arctan }[/math] by naïvely integrating term-by-term:

[math]\displaystyle{ \begin{align} \int_0^x \frac{du}{1 + u^2} &= \int_0^x \left(1 - u^2 + u^4 - u^6 + \cdots\right)du \\[5mu] &= x - \frac13 x^2 + \frac15 x^5 - \frac17 x^7 + \cdots = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}. \end{align} }[/math]

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 [math]\displaystyle{ |x| \leq 1, }[/math] [math]\displaystyle{ \arctan' }[/math] can instead be written as the finite sum,

[math]\displaystyle{ \frac{1}{1 + x^2} = 1 - x^2 + x^4 - \cdots + \bigl({-x^2}\bigr){\vphantom)}^N + \frac{\bigl({-x^2}\bigr){}^{N+1}}{1 + x^2}. }[/math]

Again integrating both sides,

[math]\displaystyle{ \int_0^x \frac{du}{1 + u^2} = \sum_{k=0}^N \frac{(-1)^kx^{2k+1}}{2k+1} + \int_0^x\frac{\bigl({-u^2}\bigr){}^{N+1}}{1 + u^2}\,du. }[/math]

In the limit as [math]\displaystyle{ N \to \infty, }[/math] the integral on the right above tends to zero when [math]\displaystyle{ |x| \leq 1, }[/math] because

[math]\displaystyle{ \begin{align} \Biggl| \int_0^x\frac{\bigl({-u^2}\bigr){}^{N+1}}{1 + u^2}\,du \,\Biggr| \,&\leq \int_0^1 \frac{u^{2N+2}}{1 + u^2}\,du \\[5mu] &\lt \int_0^1 u^{2N+2}du \,=\, \frac{1}{2N+3} \,\to\, 0. \end{align} }[/math]

Therefore,

[math]\displaystyle{ \begin{align} \arctan x = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}. \end{align} }[/math]

Convergence

The series for [math]\displaystyle{ \arctan' }[/math] and [math]\displaystyle{ \arctan }[/math] converge within the complex disk [math]\displaystyle{ |x| \lt 1 }[/math], where both functions are holomorphic. They diverge for [math]\displaystyle{ |x| \gt 1 }[/math] because when [math]\displaystyle{ x = \pm i }[/math], there is a pole:

[math]\displaystyle{ \frac1{1 + i^2} = \frac1{1 - 1} = \frac10 = \infty. }[/math]

When [math]\displaystyle{ x = \pm1, }[/math] the partial sums [math]\displaystyle{ \sum_{k=0}^n (-x^2)^k }[/math] alternate between the values [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1, }[/math] never converging to the value [math]\displaystyle{ \arctan'(\pm1) = \tfrac12. }[/math]

However, its term-by-term integral, the series for [math]\displaystyle{ \arctan, }[/math] (barely) converges when [math]\displaystyle{ x = \pm 1, }[/math] because [math]\displaystyle{ \arctan' }[/math] disagrees with its series only at the point [math]\displaystyle{ \pm1, }[/math] so the difference in integrals can be made arbitrarily small by taking sufficiently many terms:

[math]\displaystyle{ \lim_{N \to \infty} \int_0^1\biggl(\frac{1}{1 + u^2} - \sum_{k=0}^N (-u^2)^k\biggr)du = 0 }[/math]

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 [math]\displaystyle{ \tfrac14\pi. }[/math] Finding ways to get around this slow convergence has been a subject of great mathematical interest.

History

The earliest person to whom the series can be attributed with confidence is Madhava of Sangamagrama (c. 1340 – c. 1425). The original reference (as with much of Madhava'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 [math]\displaystyle{ \arctan }[/math] include Nilakantha Somayaji's Tantrasangraha (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.

See also

• List of mathematical series
• Madhava series

References

• Berggren, Lennart; Borwein, Jonathan; Borwein, Peter, eds (2004). Pi: A Source Book (3rd ed.). Springer. doi:10.1007/978-1-4757-4217-6. ISBN 978-1-4419-1915-1. 
• Gupta, Radha Charan (1973). "The Madhava–Gregory series". The Mathematics Education 7: B67–B70. 
• Horvath, Miklos (1983). "On the Leibnizian quadrature of the circle.". Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) 4: 75–83. http://ac.inf.elte.hu/Vol_004_1983/075.pdf. 
• Roy, Ranjan (1990). "The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha". Mathematics Magazine 63 (5): 291–306. doi:10.1080/0025570X.1990.11977541. https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf.