Radius of Convergence
一power series will converge only for certain values of。例如,converges for。In general, there is always an intervalin which apower
seriesconverges, and the numberis called the radius
of convergence (while the interval itself is called the interval of convergence).
The quantityis called the radius of convergence because,
in the case of a power series with complex coefficients, the values of与form anopen diskwith radius。
一power series总是converges absolutelywithin its radius of convergence. This can be seen by fixingand supposing that there exists asubsequence such that是unbounded。然后power
series does notconverge(in fact, the terms are unbounded) because it fails thelimit
test。Therefore, for与, the
power series does not converge, where

(1)


(2)

和denotes thesupremum
limit。
Conversely, suppose that。Then for
any radius与, the
terms满足

(3)

为large enough (depending on). It is sufficient
to fix a value forin between和。Because, the power
series is dominated by a convergentgeometric series.
Hence, thepower seriesconverges absolutely by thelimit comparison test。