A power series in a variable is an infinite Sum of the form

(1) |

A Conjecture of Pólya is that if a Function has a Power series with Integer Coefficients and Radius of Convergence 1, then either the Function is Rational or the Unit Circle is a natural boundary.

A generalized Power sum for , 1, ... is given by

(2) |

(3) |

For any power series, one of the following is true:

- 1. The series converges only for .
- 2. The series converges absolutely for all .
- 3. The series converges absolutely for all in some finite open interval and diverges if or . At the points and , the series may converge absolutely, converge conditionally, or diverge.

(4) |

**References**

Arfken, G. ``Power Series.'' §5.7 in *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL:
Academic Press, pp. 313-321, 1985.

Myerson, G. and van der Poorten, A. J. ``Some Problems Concerning Recurrence Sequences.'' *Amer. Math. Monthly* **102**, 698-705, 1995.

Pólya, G. *Mathematics and Plausible Reasoning, Vol. 2: Patterns of Plausible Inference.*
Princeton, NJ: Princeton University Press, p. 46, 1954.

© 1996-9

1999-05-26