The fact that a series converges if and only if the partial sum from n to m vanishes as n and m go to infinity is known as his criterion for series, and he shares credit with Hadamard for a formula which states that the radius of convergence of a power series is the reciprocal of the limsup of the k-th root of the k-th coefficient.
The fact that a series converges if and only if the partial sum from n to m vanishes as n and m go to infinity is known as his criterion for series, and he shares credit with Hadamard for a formula which states that the radius of convergence of a power series is the reciprocal of the limsup of the k-th root of the k-th coefficient. His estimate relating the maximum of an analytic function on a circle to the value of the derivative at the center provides an easy proof of Liouville's theorem, but his most important theorem in complex analysis states that if a function is analytic on a domain, the line integral over the boundary vanishes. FTP, name this mathematician who also formalized calculus with his "epsilon-delta" definition of limits.
The fact that a series converges if and only if the partial sum from n to m vanishes as n and m go to infinity is known as his criterion for series, and he shares credit with Hadamard for a formula which states that the radius of convergence of a power series is the reciprocal of the limsup of the k-th root of the k-th coefficient.
Augustin Louis {Cauchy}