## Theorem 8.3.7: Power Series

Every power series

*a*centered at_{n}(x - c)^{n}= a_{0}+ a_{1}(x-c) + a_{0}(x-c)^{2}+ ...*c*has the following properties:- The power series converges at its center, i.e. for
*x = c* - There exists an
*r*such that the series converges absolutely and uniformly for all*|x - c| p*, where*p < r*, and diverges for all*|x - c| > r*. The number*r*is called the*radius of convergence*for the power series and is given by:*r = lim inf | a*_{n}/ a_{n+1}|

*only*for*x = c*) or to be (i.e. the series converges*for all**x*).
This theorem is sometimes called **Abel's theorem on Power Series**.

Clearly the power series converges for *x = c* since then all
terms except the first reduce to zero. For the second statement, we will
simply apply the
Ratio test for series:

The series
*
a _{n} (x - c)^{n}*
converges absolutely if:

But then, taking the reciprocal:

which proves that the series converges absolutely for
*|x - c| < r*. The fact that it converges uniformly on any
closed disk centered at *c* with radius *p < r*
follows from the
Weierstrass Convergence theorem.