Bernoulli approximation to sine and cosine functions

ABSTRACT

Taylor series play a ubiquitous role in calculus courses, and their applications as approximants to functions are widely taught and used everywhere. However, it is not common to present the students with other types of approximations besides Taylor polynomials. These notes show that polynomials construed to satisfy certain boundary conditions at an interval of definition can be represented by rapidly converging Fourier series. These polynomials are shifted and re-scaled versions of the Bernoulli polynomials. From this construction, an argument to ‘invert’ the Fourier series to obtain an approximation to sines and cosines in terms of the Bernoulli polynomials is presented. We then show that approximating sines and cosines by the Bernoulli polynomials might be much better than using the truncated Taylor series, especially in problems where global proprieties are desired. These contents can be taught in advanced Calculus classes approaching series of functions and their applications.

KEYWORDS:

• Bernoulli polynomials
• Taylor polynomials
• Fourier series
• sine and cosine functions

Acknowledgments

I would like to thank Prof. Roberto da Silva for reading the manuscript and suggesting improvements, and Pedro Henrique Dario for his participation in the early stages of this work.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 An interesting expansion of trigonometric functions in terms of power series with Bernoulli numbers coefficients can be found here (Deeba & Rodriguez, 1990).

2 There is no linear function that satisfies the periodic boundary conditions.