Fun with Sign

Abstract

When I was teaching calculus recently, an exercise in the textbook [1, p. 687, qn. 3] asked students to find the Taylor series of cosx centred at π/4. My students were able to find the seriesbut wondered how to take care of the sign changes, in order to express the series in I.-notation. They were quite puzzled after finding that the answer given in [I] isThe exponent n (n + 1)/2 seemed a rabbit out of the hat! This article is the result of trying to answer their question satisfactorily, and to have some serious fun. Readers are invited to try two teasers first before reading for their solutions.