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- 1.
Most of Cauchy’s contemporaries continued to treat infinite series as if they were very large finite series that carried with them the same properties. This tradition dated back through the days of Euler to at least the time of Isaac Newton. As an example, all through the 18th century and into Cauchy’s days an infinite power series was generally treated as though it was simply a large polynomial. However, Cauchy now demands the convergence of an infinite series be demonstrated before any properties can be assigned.
- 2.
Properties of the common geometric series are covered thoroughly in Chapter VI of Cauchy’s Cours d’analyse. Even though this earlier text was never officially used in practice, one must assume his students would have been expected to be quite familiar with this fundamental series.
- 3.
Cauchy seems to be taking for granted that absolute convergence of a series implies its con-vergence. Absolute convergence is not a concept Cauchy ever defines specifically within Calcul infinitésimal, but this current assumption is one he essentially proves in his Cours d’Analyse Chap-ter VI, §III.
- 4.
Cauchy is stating what we now know as the Root Convergence Test. He has also demonstrated similar versions of this theorem in his Cour d’analyse Chapter VI, §II, Theorem I and Chapter VI, §III, Theorem I.
- 5.
This theorem is today known as the Ratio Convergence Test. Cauchy has stated similar versions of this result in his Cours d’analyse Chapter VI, §II, Theorem II and Chapter VI, §III, Theorem II.
- 6.
Cauchy is pointing out his Theorem I result (the Root Convergence Test) is stronger and more general than his Theorem II result (the Ratio Convergence Test).
- 7.
This theorem has been corrected. Both the 1823 and 1899 editions incorrectly define \(\rho _n\) to be \(F^{(n)}(0).\) The error must clearly be a typographical omission dating back to 1823, as Cauchy knew better. He even demonstrates a related result in his Cours d’analyse Chapter VI, §IV, Theorem II, Corollary IV from 1821.
- 8.
Cauchy is presenting here what today we would refer to as the radius of convergence of the Maclaurin series for each of these functions.
- 9.
Cauchy is likely referring to the work of Lagrange.
- 10.
The example to follow demonstrates the Taylor series for the two functions, \(H(x)=0 \) and \(F(x)=e^{-\left( \frac{1}{x}\right) ^2} \) are identical, even though the functions themselves are certainly not. This example will also illustrate the Taylor series for \(F(x)=e^{-\left( \frac{1}{x}\right) ^2}\) does not accurately represent the original function, as the series will suggest F(x) is identically zero for all values of x, when the function itself is never zero, as Cauchy points out. With this single example, Cauchy clearly shows that not every function can be written as a convergent Taylor series and that a Taylor series does not uniquely identify a function.
- 11.
This second counterexample of \(F(x)=e^{-x^2}+e^{-\left( \frac{1}{x}\right) ^2}\) produces a Taylor series identical to that of the distinct function \(G(x)=e^{-x^2},\) demonstrating once again a Taylor series representation does not uniquely identify a function.
As discussed earlier, one of Cauchy’s early contemporaries and mentors, Joseph-Louis Lagrange, had previously proposed (years before) that the derivative of a function be defined as the coef-ficient of the linear term of its Taylor series expansion. This theory had even been a part of the École Polytechnique curriculum in its early years. Cauchy uses these last two examples to clearly illustrate this theory is not valid. Among other problems, such as the fact some Taylor series do not converge, he shows different functions can produce the same series. These last two examples play a pivotal role in the demolishment of Lagrange’s foundational assumptions of functions and the Lagrangean theory of the calculus. Ironically, even though Cauchy helps dismantle Lagrange’s theory, one of Cauchy’s main references while developing his own theory was Lagrange’s Théorie des fonctions analytiques published in installments beginning in 1797. In this latter text, Lagrange presents his work without any diagrams; a practice Cauchy continued.
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Cates, D.M. (2019). RULES ON THE CONVERGENCE OF SERIES. APPLICATION OF THESE RULES TO THE SERIES OF MACLAURIN.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_38
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