Abstract
We call a series an indefinite sequence of terms
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- 1.
Cauchy is careful with his wording here. He is defining s, the sum of a convergent infinite series, as the limit of the partial sums, \(s_n, \) as n gets large. Specifically, he is not defining s to be any of the \(s_n\)’s. Cauchy defines the value of s as a limit, when the limit exists.
- 2.
The concept of a divergent series having no sum, in other words having no meaning whatsoever, was not as obvious an idea at the time of Cauchy as we see it today. Many mathematicians up to his day had avoided any such conclusion and had considered divergent series just as valid as convergent series. There was a belief, even into the 19th century, that these divergent series would turn out to be justified in some manner, and so, there was a great deal of reluctance to exclude them from a general theoretical analysis of infinite series. Cauchy stands strong in this regard and makes the statement that unless an infinite series converges to a sum, none of his analysis to follow applies. This is a bold leap for Cauchy to make in 1823, one he outlines at the beginning of this text in his FOREWORD.
- 3.
The 1899 reprint has \(s-s_n=r_n. \ \)
- 4.
A sufficient condition for convergence (as will be shown in a subsequent example). Cauchy is quite aware that in general it is not enough for the ending terms of a sequence to be approaching zero to guarantee the corresponding series converges. The classic counterexample illustrating this possibility is the harmonic series, which diverges despite its terms becoming smaller and smaller. What is different in Cauchy’s case here is that by knowing the remainder of the series is becoming smaller, it is known the sum of all the ending terms combined is also approaching zero. This latter condition is enough to guarantee convergence as Cauchy correctly argues.
- 5.
The 1899 text has a clear misprint in equation (6). It reads, \( F(x)=F(0)+\frac{1}{x}F^{\prime }(0)+\frac{x^2}{1\cdot 2}F^{\prime \prime }(0)+\frac{x^3}{1\cdot 2\cdot 3}F^{\prime \prime \prime }(0)+\cdots . \)
- 6.
The Mean Value Theorem for Definite Integrals.
- 7.
Known today as the remainder in Cauchy form.
- 8.
In today’s nomenclature, as long as the \(n^{th}\) derivative is bounded. Using this condition for \(F^{(n)}(z)\), to verify the expression in (14) tends to zero Cauchy only needs to show the preceding term, \(\frac{(x-z)^{n-1}}{1\cdot 2\cdot 3 \cdots (n-1)}\), vanishes as n gets large. It would be interesting to be sitting in a desk at the back of Cauchy’s École Polytechnique classroom in 1823, given the descriptive set of subsequent steps he provides as clues here, to have him explain his demonstration of these valid claims.
- 9.
This theorem, as well as the Maclaurin series, are named after Colin Maclaurin (1698–1746) of Scotland, one of Isaac Newton’s most successful followers.
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Cates, D.M. (2019). THEOREMS OF TAYLOR AND OF MACLAURIN. EXTENSION OF THESE THEOREMS TO FUNCTIONS OF SEVERAL VARIABLES.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_37
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