Complex Power Series
Convergence and Absolute Convergence
∞
X
•
Cn CONVerges iff
∗
lim
N
X
N →∞
Cn
∗
exists (finite).
• If Cn = xn + iyn , then
•
∞
X
∞
X
Cn CONVerges iff
∗
Cn CONVerges ABSsolutely iff
∞
X
∗
∞
X
xn CONVerges and
∞
X
∗
yn CONVerg
∗
|Cn | converges or
∗
lim
N →∞
N
X
|Cn |
∗
exists (finite).
• If Cn = xn + iyn , then
∞
X
Cn CONVerges ABSolutely iff
∗
∞
X
∗
xn and
∞
X
yn CONVe
∗
ABSolutely.
∞
N
X
X
•
|Cn | converges iff the sequence
|Cn | is bounded.
∗
∗
Comparison Test for Absolute Convergence
• If
|Cn | ≤ Bn ,
then
0≤
∞
X
|Cn | ≤
∞
X
∗
So that if the series
∞
X
Bn CONVerges, the
∗
Bn .
∗
∞
X
An CONVerges ABSolutely. Moreo
∗
there is the obvious error estimate
∞
∞
X
X
Cn ≤
|Cn |
The Geometric Series
We consider the geometric series
∞
X
zn.
0
A bare hands calculation shows that
(1 − z)
N
X
z n = (1 − z) 1 + z + . . . + z N
0
= 1 − z N +1 .
so that
∞
X
z
n
1 for |z| < 1,
CONVerges ABSolutely to 1−z
DIVerges for |z| ≥ 1.
0
For |z| < r < 1, there is the error estimate
∞
∞
N +1
X X
|z|
n
|z| =
zn ≤
1 − |z|
N +1
N +1
z N +1 rN +1
≤ r
1−r
N +1
r
≤
.
1−r
Ratio Test for ABSolute CONVergence
∞
X
For the series
CN , suppose that
∗
Cn+1 = L.
lim n→∞ Cn • If 0 ≤ L < 1, the series
∞
X
CN CONVerges ABSolutely.
∗
Why? Choose an r, L < r < 1. For n sufficiently large, |Cn+1 | < r |Cn | and for N la
enough,
∞
X
N
|Cn | ≤
∞
X
|CN | rn−N
N
= |CN |
1
.
• If L = 1, we are not sure – additional information is needed to decide DIVergence
CONVergence and/or ABS0lute CONVergence.
Power Series, Radius of Convergence, and Circle/Disk of Convergence
∞
X
• If the power series
an z n , converges for a nonzero z = z0 , then for all z, |z| < |z0 |,
n=0
power series CONVerges ABSolutely.
Why? We have that limn→∞ |an z0n | = 0. Then we can compare
∞
X
∞
X
n
z
|an z | =
z0 N +1
N +1
X
∞
n
≤ max |an z0 |
n
|an z0n | n≥N
= o (1)
N +1
n
z
z0 θN +1
,
1−θ
where o (1) means limN →∞ o (1) = 0, and θ = zz0 .
Thus the convergence of the series at a nonzero z0 forces the absolute convergence of
series in the entire open disk centered at 0 with radius |z0 |.
∞
X
• For a power series
an z n , there is a number R, 0 ≤ R ≤ ∞ for which
n=0
∞
X
an z
n
n=0
CONVerges ABSolutely for |z| < R,
DIVerges for |z| > R.
The number R is called the radius of convergence of the power series. R can often
determined by the Ratio Test.
• If f (z) is represented by a convergent power series for |z| < R, then f (z) is an analy
function in the region |z| < R and its derivative is represented by the convergent ser
∞
X
nan z n−1 , |z| < R.
n=1
Thus the power series for f 0 has radius of convergence at least R, and the forma
differentiated series converges to the analytic function f 0 (z). Within the open disk
convergence, it follows that function represented by a power series has derivatives of
orders which are represented by the series differentiated term by term.
then
0
f (z) =
∞
X
n=1
and1
Z
0
z
nan z
n−1
=
∞
X
(n + 1)an+1 xn , |x| < R,
n=0
∞
∞
X
an n+1 X an−1 n
f (ζ) dζ =
z
=
z , |z| < R
n+1
n
n=0
n=1