INTEGRATION BY SERIES

Dennis M. Cates ()

Chapter Chapter 40 in Cauchy's Calcul Infinitésimal, 2019, pp 221-226 from Springer

Abstract: Abstract Consider a series whose different terms are functions of the variable x, that remain continuous between the limits $$x=x_0, $$ $$x=X. \ $$ If, after having multiplied these same terms by dx, we integrate between the limits in question, we will obtain a new series composed of the definite integrals $$\begin{aligned} \int _{x_0}^{X}{u_0 \, dx}, \ \ \ \ \ \int _{x_0}^{X}{u_1 \, dx}, \ \ \ \ \ \ \int _{x_0}^{X}{u_2 \, dx}, \ \ \ \ \ \int _{x_0}^{X}{u_3 \, dx}, \ \ \ \ \ \dots , \ \ \ \ \ \int _{x_0}^{X}{u_n \, dx}, \ \ \ \ \ \dots . \end{aligned}$$ By comparing this new series to the first, we will establish without difficulty the theorem that we now state.

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_40

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