Output Data Structures

Abstract

Like his contemporaries, Euler was happy to use divergent series for the purposes of calculation. He was able to avoid errors firstly because his instincts were sure, but also because the functions he considered were relatively simple. For the mathematicians of Euler’s day a function was something given explicitly by a formula, as indeed it is for us. Later, Abel and Cauchy began to regard a function as a process acting on numbers, and thereby came to consider much more general types of function. When the functions were no longer given so explicitly much more care was needed with the logic. Whether or not this lay behind their reasoning, Abel and Cauchy sought to impose a ban on the use of divergent series. From the point of view of pure mathematics at that time this was undoubtedly justified, but divergent series continued to give surprisingly accurate results, and continued to have their adherents.