A series of surprises
O'Farrell, Anthony G. (1990) A series of surprises. Irish Mathematics Teachers' Association Newsletter (73). pp. 28-33.
Riemann’s theorem on conditionally–convergent series surprises a lot of people. Some people react to it by retreating to the view that only absolute convergence deserves to be taken seriously. I go the other way. What I take from it is that the ordinary notion of the convergence of a series is not so sacred, after all. That notion relates to one particular way of adding up a series, one of many. Depending on the circumstances, one of the other ways may be more appropriate or interesting. The idea that a series should add up in the usual way is just a prejudice. For instance, with the Fourier series of a continuous function, it is a fact that the series often fails to converge in the ordinary way, but we know that the Cesaro means always converge uniformly to the function.
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