14—Complex Variables 425
The structure of a Laurent series is such that it will converge in an annulus. Examine the absolute
convergence of such a series.
∑∞
−∞
∣
∣akzk
∣
∣=
∑−^1
−∞
∣
∣akzk
∣
∣+
∑∞
0
∣
∣akzk
∣
∣
The ratio test on the second sum is
if for large enoughk,
|ak+1||z|k+1
|ak||z|k
=
|ak+1|
|ak|
|z|≤x < 1 then the series converges.
This defines the largest value of|z|for which the sum of positive powers converges.
If|ak+1|/|ak|has a limit then |z|max= lim
|ak|
|ak+1|
Do the same analysis for the series of negative powers, applying the ratio test.
if for large enoughnegativek,
|ak− 1 ||z|k−^1
|ak||z|k
=
|ak− 1 |
|ak|
1
|z|
≤x < 1 then the series converges.
This defines thesmallestvalue of|z|for which the sum of negative powers converges.
If|ak− 1 |/|ak|has a limit ask→−∞then |z|min= lim
|ak− 1 |
|ak|
If|z|min < |z|max then there is a range ofz for which the series converges absolutely (and so of course it
converges).
|z|min<|z|<|z|max an annulus
|z|min
|z|max
If either of these series of positive or negative powers is finite, terminating in a polynomial, then respectively
|z|max=∞or|z|min= 0.