Power Series

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

$$\sum_n^\infty a_iz^i,$$ (1)

where $n\geq 0$ and $a_i$ are Integers, Real Numbers, Complex Numbers, or any other quantities of a given type.

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 $a(h)$ for $h=0$, 1, ... is given by

$$a(h)=\sum_{i=1}^m A_i(h){\alpha_i}^h,$$ (2)

with distinct Nonzero Roots $\alpha_i$, Coefficients $A_i(h)$ which are Polynomials of degree $n_i-1$ for Positive Integers $n_i$, and $i\in[1,m]$. The generalized Power sum has order

$$n\equiv \sum_{i=m}^m n_i.$$ (3)

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

1. The series converges only for $x = 0$.

2. The series converges absolutely for all $x$.

3. The series converges absolutely for all $x$ in some finite open interval $(-R,R)$ and diverges if $x < -R$ or $x > R$. At the points $x = R$ and $x = -R$, the series may converge absolutely, converge conditionally, or diverge.

To determine the interval of convergence, apply the Ratio Test for Absolute Convergence and solve for $x$. A Power series may be differentiated or integrated within the interval of convergence. Convergent power series may be multiplied and divided (if there is no division by zero).

$$\sum_{k=1}^\infty k^{-p}.$$ (4)

Converges if $p > 1$ and Diverges if $0<p\leq 1$.

See also Binomial Series, Convergence Tests, Laurent Series, Maclaurin Series, Multinomial Series, p-Series, Polynomial, Power Set, Quotient-Difference Algorithm, Recurrence Sequence, Series, Series Reversion, Taylor Series

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.