Isaac Newton introduced differential calculus in 1666, although he did not publish his discoveries until later. Half a century later, his fellow countryman Brook Taylor attempted to approximate functions near 0 and claimed that, for x close to 0, f(x) = f (0) + x f '(0) + x2f ''(0) / 2… At the time, no one bothered to specify that f had to be differentiable several times, or to explain what the ellipsis meant: did it represent a negligible quantity, or did adding more and more terms bring the expression closer to equality?
The theory of power series, developed subsequently, made it possible to give sufficient conditions for obtaining such an equality by extending the sum to infinity (the series is then said to converge).
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From Taylor to de Moivre -------------------------
We know that the exponential function is its own derivative. It follows that it is infinitely differentiable and that, for every x, its nth derivative is f (n)(x) = exp (x). Suppose—as is indeed the case—that Taylor was right for this particular function. Then