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.

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