Why was Lagrange’s assumption about power series later found to be false?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Joseph-Louis Lagrange’s assumption that every function could be represented by a power series expansion was later proven false by Augustin-Louis Cauchy during the 1820s.1
Lagrange had attempted to build the foundations of calculus entirely on algebraic analysis, defining the derivative through the coefficients of a Taylor series. However, this approach lacked the necessary rigor to account for functions that do not behave in this manner. Cauchy rejected this purely algebraic framework, demonstrating that the assumption of universal power series representation was incorrect.12
Instead, Cauchy established a more rigorous foundation for calculus by prioritizing the concepts of continuity and limit over algebraic expansion. This shift allowed mathematicians to define differentiable and integrable functions without relying on the unproven and ultimately flawed premise that all functions are analytic (expressible as power series).1