Chebyshev polynomials boost the π-series convergence

Abstract We present an interesting application of the solution to the simple harmonic oscillator (SHO) that can serve as a computation of $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> . We begin with a review of a compact teaching strategy for solving its equation of motion through integration in a general physics course, where many students face difficulties with conventional methods for solving differential equations. This integration approach leads to the arcsine function, the inverse of the sine function, ultimately providing the solution to the SHO. We investigate various series for approximating $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> , focusing on the arcsine series and their difference in convergence speed. We begin with Newton’s arcsine series for $$\pi =2\arcsin 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>arcsin</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We then explore a series based on powers of $$\sin \frac{\pi }{2^{k+1}} \ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>sin</mml:mo> <mml:mfrac> <mml:mi>π</mml:mi> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mfrac> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , where k is a large positive integer and the sine term is computed using nested radicals through half-angle formulas, resembling Viète’s formula. The small sine term acts as a power-counting parameter, making the series better convergent to $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> with reliable error estimation. We extend this approach to a fractional-angle method, generalizing the factor from 1/2 to $$1/{p'}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> for a prime number $$p'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> , by employing Chebyshev polynomials of the second kind, which commonly arise in physics problems. This leads to a series involving powers of $$\sin \frac{\pi }{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>sin</mml:mo> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mi>p</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> , where p is an arbitrary integer expressed as a product of prime factors, further enhancing convergence with a smaller power-counting parameter. The power counting allows us to identify significant terms in the Chebyshev polynomials and to truncate numerically insignificant contributions that optimize and simplify the computation of the sine term. Our novel strategies are pedagogical and suitable for advanced physics undergraduates, enabling them to approximate $$\pi$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> with high accuracy using techniques covered in physics courses.