A student usually first meets power series through an infinite geometric progression, having previously considered finite geometric progressions. In this note we consider a variation of this introductory material which involves the Fibonacci numbers. This necessarily poses various questions, e.g. ’When does the series converge and, if so, what is the sum?’. However, there is one further intriguing question that is natural to ask, and this leads to some interesting mathematics. All of this is appropriate for sixth formers, either for classroom discussion or as an exercise.