In this paper we investigate two problems concerning the theory of power series in 18th-century mathematics: the development of a given function into a power series and the inverse problem, the return from a given power series to the function of which this power series is the development. The way of conceiving and solving these problems closely depended on the notion of function and in particular on the conception of a series as the result of a formal transformation of a function. After describing the procedures considered acceptable by 18th-century mathematicians, we examine in detail the different strategies—both direct and inverse, that is, synthetic and analytical—they employed to solve these problems.
Marco Panza, Giovanni Ferraro. Developing into series and returning from series: A note on the foundations of eighteenth-century analysis. Historia Mathematica, 2003, 30, pp.17-46. https://doi.org/10.1016/S0315-0860(02)00017-4
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