text: Terms with indices that are powers of two are used in this mathematician's namesake condensation test to check for convergence of a series.
full_question: Terms with indices that are powers of two are used in this mathematician's namesake condensation test to check for convergence of a series. With Hadamard, this mathematician names a formula that calculates the radius of convergence of a power series. He's not Lagrange, but he names a theorem stating that if a prime p divides the order of a group, then the group contains an element with order p. Partial differential equations that give a necessary and sufficient condition for a complex function to be holomorphic are named for him and Riemann. Sequences named for this man, by definition, converge inside complete metric spaces. For 10 points, name this French mathematician who names an inequality relating the dot product of two vectors and their magnitudes along with Hermann Schwarz.
first_sentence: Terms with indices that are powers of two are used in this mathematician's namesake condensation test to check for convergence of a series.
raw_answer: Augustin-Louis Cauchy