text: Pólya's extension of Carlson's theorem guarantees that some of these functions have the unit circle as their boundaries, or are rational.
full_question: Pólya's extension of Carlson's theorem guarantees that some of these functions have the unit circle as their boundaries, or are rational. Padé approximants are the ratio of two of these functions. A critical parameter of these functions is the limit superior as n goes to infinity of the value at n, raised to one over n, in the Cauchy- Hadamard theorem. By definition, a function is analytic if it can be written as one of these functions locally. The root test and ratio tests are applied to these functions to determine their radii of convergence. The simplest of these functions is equivalent to one over one minus x. For 10 points, name these infinite series containing all the exponents of a variable.
first_sentence: Pólya's extension of Carlson's theorem guarantees that some of these functions have the unit circle as their boundaries, or are rational.
raw_answer: power series [prompt on series; anti-prompt on Taylor series or Maclaurin series]