The Basel Problem and Summing Rational Functions over Integers

Authors

Pranjal Jain, Indian Institute of Science Education and Research, PuneFollow

Abstract

We provide a general method to evaluate convergent sums of the form ∑_{k∈Z} R(k) where R is a rational function with complex coefficients. The method is entirely elementary and does not require any calculus beyond some standard limits and convergence criteria. It is inspired by a geometric solution to the famous Basel Problem given by Wästlund (2010), so we begin by demonstrating the method on the Basel Problem to serve as a pilot application. We conclude by applying our ideas to prove Euler’s factorisation for sin x which he originally used to solve the Basel Problem.

Author Bio

Pranjal Jain is pursuing a BS-MS dual degree at IISER, Pune. He was inspired to pursue this research by a related YouTube video published by the channel 3blue1brown. In his spare time he enjoys jogging, music and playing the guitar.

Faculty Sponsor

Dr. Chandrasheel Bhagwat

Recommended Citation

Jain, Pranjal (2024) "The Basel Problem and Summing Rational Functions over Integers," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 1, Article 4.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss1/4