Arctangent series

File:Arctan and its derivative.png

If $y\; =\; \backslash arctan\; x$ then $\backslash tan\; y\; =\; x.$ The derivative is

:$$

\frac{dx}{dy} = \sec^2 y = 1 + \tan^2 y.

Taking the reciprocal,

:$$

\frac{dy}{dx} = \frac{1}{1 + \tan^2 y} = \frac{1}{1 + x^2}.

This sometimes is used as a definition of the arctangent:

:$\backslash arctan\; x\; =\; \backslash int\_0^x\backslash frac\{du\}\{1\; +\; u^2\}.$

The Maclaurin series for $x\; \backslash mapsto\; \backslash arctan\text{'}\; x\; =\; 1\; \backslash big/\; \backslash left(1\; +\; x^2\backslash right)$ is a geometric series:

:$\backslash frac\{1\}\{1\; +\; x^2\}\; =\; 1\; -\; x^2\; +\; x^4\; -\; x^6\; +\; \backslash cdots\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash bigl(\{-x^2\}\backslash bigr)\{\backslash vphantom)\}^k.$

One can find the Maclaurin series for $\backslash arctan$ by naïvely integrating term-by-term:

:$\backslash begin\{align\}$

\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^3 + \frac15 x^5 - \frac17 x^7 + \cdots

= \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}.

\end{align}

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 $|x|\; \backslash leq\; 1,$ $\backslash arctan\text{'}$ can instead be written as the finite sum,

{{cite journal |last=Shirali |first=Shailesh A. |year=1997 |title=Nīlakaṇṭha, Euler and {{mvar|π}} |journal=Resonance |volume=2 |issue=5 |pages=29–43 |doi=10.1007/BF02838013 |s2cid=121433151 |url=https://www.ias.ac.in/article/fulltext/reso/002/05/0029-0043 }} Also see the erratum: {{cite journal |last=Shirali |first=Shailesh A. |year=1997 |title=Addendum to 'Nīlakaṇṭha, Euler and {{mvar|π}}' |journal=Resonance |volume=2 |issue=11 |page=112 |doi=10.1007/BF02862651 |doi-access=free }}

:$$

\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}.

Again integrating both sides,

:$$

\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.

In the limit as $N\; \backslash to\; \backslash infty,$ the integral on the right above tends to zero when $|x|\; \backslash leq\; 1,$ because

:$\backslash begin\{align\}$

\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]

&< \int_0^1 u^{2N+2}du

\,=\, \frac{1}{2N+3} \,\to\, 0.

\end{align}

Therefore,

:$\backslash begin\{align\}$

\arctan x = \sum_{k=0}^\infty \frac{(-1)^kx^{2k+1}}{2k+1}.

\end{align}

The series for $\backslash arctan\text{'}$ and $\backslash arctan$ converge within the complex disk , where both functions are holomorphic. They diverge for $|x|\; >\; 1$ because when $x\; =\; \backslash pm\; i$, there is a pole:

:$\backslash frac1\{1\; +\; i^2\}\; =\; \backslash frac1\{1\; -\; 1\}\; =\; \backslash frac10\; =\; \backslash infty.$

When $x\; =\; \backslash pm1,$ the partial sums $\backslash sum\_\{k=0\}^n\; (-x^2)^k$ alternate between the values $0$ and $1,$ never converging to the value $\backslash arctan\text{'}(\backslash pm1)\; =\; \backslash tfrac12.$

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

:$\backslash lim\_\{N\; \backslash to\; \backslash infty\}\; \backslash int\_0^1\backslash biggl(\backslash frac\{1\}\{1\; +\; u^2\}\; -\; \backslash sum\_\{k=0\}^N\; (-u^2)^k\backslash biggr)du\; =\; 0$

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 $\backslash tfrac14\backslash pi.$ Finding ways to get around this slow convergence has been a subject of great mathematical interest.

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 {{nobr|1=

{{cite book |last=Roy |first=Ranjan |year=2021 |orig-year=1st ed. 2011 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |pages=215–216, 219–220}}

{{pb}}{{cite web |last=Sandifer |first=Ed |year=2009 |title=Estimating π |website=How Euler Did It |url=http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf }} Reprinted in {{cite book |last=Sandifer |first=Ed |display-authors=0 |year=2014 |title=How Euler Did Even More |pages=109–118 |publisher=Mathematical Association of America}}

{{pb}}{{cite book |last=Newton |first=Isaac |authorlink=Isaac Newton |year=1971 |editor-last=Whiteside |editor-first=Derek Thomas |editor-link=Tom Whiteside |title=The Mathematical Papers of Isaac Newton |volume=4, 1674–1684 |publisher=Cambridge University Press |pages=526–653 }}

{{pb}} {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |year=1755 |title=Institutiones Calculi Differentialis |publisher=Academiae Imperialis Scientiarium Petropolitanae |lang=la |url=https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/ |at=[https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/318 §2.2.30 {{pgs|318}}] }} [https://scholarlycommons.pacific.edu/euler-works/212/ E 212]. Chapters 1–9 translated by John D. Blanton (2000) Foundations of Differential Calculus. Springer. Later translated by Ian Bruce (2011). [http://17centurymaths.com/contents/differentialcalculus.htm Euler's Institutionum Calculi Differentialis]. 17centurymaths.com. ([http://17centurymaths.com/contents/euler/diffcal/part2ch2.pdf English translation of §2.2])

{{pb}}{{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1798 |orig-year=written 1779 |title=Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae |journal=Nova Acta Academiae Scientiarum Petropolitinae |volume=11 |pages=133–149, 167–168 |url=https://archive.org/details/novaactaacademia11petr/page/133 |id=[https://scholarlycommons.pacific.edu/euler-works/705/ E 705] }}

{{pb}}{{citation |author=Hwang Chien-Lih |year=2005 |title=An elementary derivation of Euler's series for the arctangent function |journal=The Mathematical Gazette |volume=89 |issue=516 |pages=469–470 |doi=10.1017/S0025557200178404 }}}}

:$\backslash begin\{align\}$

\arctan x

&= \frac {x} {1 + x^2}

\sum_{n=0}^\infty \prod_{k=1}^n \frac{2k}{2k+1} \, \frac{x^2}{1 + x^2} \\[10mu]

&= \frac {x} {1 + x^2} + \frac23 \frac {x^3} {(1 + x^2)\vphantom{l}^2} + \frac{2\cdot 4}{3 \cdot 5} \frac {x^5} {(1 + x^2)\vphantom{l}^3}

+ \frac{2\cdot4\cdot6}{3\cdot5\cdot7} \frac {x^7} {(1 + x^2)\vphantom{l}^4} + \cdots \\[10mu]

&= C(x)\left(

S(x) + \frac23S(x)^3 + \frac{2\cdot 4}{3 \cdot 5}S(x)^5

+ \frac{2\cdot4\cdot6}{3\cdot5\cdot7}S(x)^7 + \cdots

\right),

\end{align}

where $\backslash vphantom\backslash Big|\; C(x)\; =\; 1\; \backslash big/\; \backslash !\backslash sqrt\{1\; +\; x^2\}\; =\; \{\}$$\backslash cos(\backslash arctan\; x)$ and $\backslash vphantom\backslash Big|\; S(x)\; =\; x\; \backslash big/\; \backslash !\backslash sqrt\{1\; +\; x^2\}\; =\; \{\}$$\backslash sin(\backslash arctan\; x).$

Each term of this modified series is a rational function with its poles at $x\; =\; \backslash pm\; i$ 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.

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 $\backslash arctan$ include Nīlakaṇṭha Somayāji's Tantrasaṅgraha (c. 1500),{{cite web|url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf |title=Tantrasamgraha with English translation |editor=K.V. Sarma |others=Translated by V.S. Narasimhan |publisher=Indian National Academy of Science |pages=48 |language=Sanskrit, English |access-date=17 January 2010 |url-status=dead |archive-url=https://web.archive.org/web/20120309014402/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf |archive-date=9 March 2012 }}Tantrasamgraha, ed. K.V. Sarma, trans. V. S. Narasimhan in the Indian Journal of History of Science, issue starting Vol. 33, No. 1 of March 1998

Jyeṣṭhadeva's Yuktibhāṣā (c. 1530),{{cite web| editor=K. V. Sarma & S Hariharan| work=Yuktibhāṣā of Jyeṣṭhadeva| url=http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf| title=A book on rationales in Indian Mathematics and Astronomy—An analytic appraisal| access-date=2006-07-09|archive-url = https://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf |archive-date = 28 September 2006}} and the Yukti-dipika commentary by Sankara Variyar, where it is given in verses 2.206 – 2.209.{{Cite book|last=C.K. Raju|title=Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16 c. CE|publisher=Centre for Studies in Civilisation|location=New Delhi|year=2007|url=https://books.google.com/books?id=jza_cNJM6fAC|series=History of Science, Philosophy and Culture in Indian Civilisation|volume=X Part 4|page=231|isbn=978-81-317-0871-2}}

• List of mathematical series
• Mādhava series

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Category:Series (mathematics)