Series part 3

Question 1

when does a seires of complex numbers converge absolutely

Answer 1

series of abs vals of those numbers convergerges

Question 2

what is the circle of convergence of a seires

Answer 2

The greatest circle centered at z0 such that power series converges at each point inside

Question 3

power seires conv outside circle of conv

Answer 3

No
A power series does not converge at any point
outside its circle of convergence

Question 4

Uniform convergence

Answer 4

Let A ⊆ C and let Sn, S : A → C be functions (n ∈ N). Then (Sn) converges uniformly to S on A if for every $\epsilon > 0 \exists N \in N$ st if $n ≥ N$ then
$|Sn(z) − S(z)| < \epsilon \forall z \in A$.

Question 5

uniform conv of a poweries on a set A

Answer 5

A power series converges uniformly on a set A
in C if its sequence of partial sums converges
uniformly on A.

Question 6

uniform conv of power seires on closed subdisks of the disk of convergence

Answer 6

if z1 is a point inside the circle of convergence |z−z0| = R of a power series $\sum_{n=0}^\infty a_n (z−z_0)^n$, then this series converges uniformly on the closed disk $|z − z_0| ≤ R_1$ where $R_1 = |z_1 − z_0|$.

Question 7

Cont of sums of power seires

when is sum of series S continuous on st A

Answer 7

Let A ⊆ C and let Sn, S : A → C be functions
(n ∈ N) such that each Sn is continuous on
A and (Sn) converges uniformly to S on A.
Then S is continuous on A.

Question 8

power series cont function under what cond

Answer 8

A power series represents a
continuous function at each point inside its
circle of convergence |z − z0| = R.

Question 9

Corallary integrals and cont functions on contours

Answer 9

Let (fn) be a sequence of complex valued
functions which are continuous on a contour
C and let f be a complex valued function
defined on C.

If (fn) converges uniformly
to f on C, then
$(\int_C f_n(z) dz)$ converges to $\int_C f(z) dz$