If Cn = xn + i yn, then
∑*∞Cn CONVerges iff
∑*∞xn CONVerges and
∑*∞yn CONVerges.
•
∑*∞Cn CONVerges ABSsolutely iff
∑*∞|Cn| converges or
lim
N → ∞
N ∑ *
|Cn|
exists (finite).
•
If Cn = xn + i yn, then
∑*∞Cn CONVerges ABSolutely iff
∑*∞xn and
∑*∞yn CONVerge ABSolutely.
•
∑*∞|Cn| converges iff the sequence ∑*N |Cn| is bounded.
Comparison Test for Absolute Convergence
•
If
|Cn| ≤ Bn,
then
0 ≤
∞ ∑ *
|Cn| ≤
∞ ∑ *
Bn.
So that if the series ∑*∞Bn CONVerges, the
∑*∞An CONVerges ABSolutely. Moreover there is the
obvious error estimate
⎢ ⎢
∞ ∑ N+1
Cn
⎢ ⎢
≤
∞ ∑ N+1
|Cn|
≤
∞ ∑ N+1
Bn
The Geometric Series
We consider the geometric series
∞ ∑ 0
zn.
A bare hands calculation shows that
(1 − z)
N ∑ 0
zn
=(1 − z) (1 + z + …+ zN)
= 1 − zN+1.
so that
∞ ∑ 0
zn
⎧ ⎪ ⎨
⎪ ⎩
CONVergesABSolutelyto
1
1−z
for |z| < 1,
DIVergesfor |z| ≥ 1.
For |z| < r < 1, there is the error estimate
⎢ ⎢
∞ ∑ N+1
zn
⎢ ⎢
≤
∞ ∑ N+1
|z|n
=
|z|N+1
1 − |z|
≤
⎢ ⎢
z
r
⎢ ⎢
N+1
rN+1
1 − r
≤
rN+1
1 − r
.
Ratio Test for ABSolute CONVergence
For the series ∑*∞CN, suppose that
lim
n → ∞
⎢ ⎢
Cn+1
Cn
⎢ ⎢
= L.
•
If 0 ≤ L < 1, the series ∑*∞CN
CONVerges ABSolutely.
Why? Choose an r, L < r < 1. For n sufficiently large,
|Cn+1| < r |Cn| and for N large enough,
∞ ∑ N
|Cn|
≤
∞ ∑ N
|CN|rn − N
= |CN|
1
1−r
.
•
If 1 < L ≤ ∞, the series ∑*∞CN
DIVerges.
•
If L = 1, we are not sure - additional
information is needed to decide DIVergence or CONVergence and/or ABS0lute
CONVergence.
Power Series, Radius of Convergence, and Circle/Disk of Convergence
•
If the power series ∑n=0∞ an zn, converges for a nonzero z = z0, then for all z, |z| < |z0|, the power series
CONVerges ABSolutely.
Why? We have that limn → ∞ |an z0n| = 0. Then
we can compare
∞ ∑ N+1
|an zn|
=
∞ ∑ N+1
|an z0n|
⎢ ⎢
z
z0
⎢ ⎢
n
≤
⎛ ⎝
max
n ≥ N
|an z0n|
⎞ ⎠
∞ ∑ N+1
⎢ ⎢
z
z0
⎢ ⎢
n
= o(1)
θN+1
1 − θ
,
where o(1) means limN → ∞ o(1) = 0, and
θ = |[z/(z0)]|.
Thus the convergence of the series at a nonzero z0 forces the
absolute convergence of the series in the entire open disk
centered at 0 with radius |z0|.
•
For a power series ∑n=0∞ an zn, there is a number
R, 0 ≤ R ≤ ∞ for which
∞ ∑ n=0
an zn
⎧ ⎪ ⎨
⎪ ⎩
CONVergesABSolutelyfor |z| < R,
DIVergesfor |z| > R.
The number R is called the radius of convergence of
the power series. R can often be determined by the Ratio Test.
•
If f(z) is represented by a convergent power series for
|z| < R, then f(z) is an analytic function in the
region |z| < R and its derivative is represented by the
convergent series ∑n=1∞ n an zn−1, |z| < R.
Thus the power series for f′ has radius of convergence
at least R, and the formally differentiated series converges
to the analytic function f′(z). Within the open disk of convergence,
it follows that function represented by a power series has
derivatives of all orders which are represented by the series
differentiated term by term.