Notions of Convergence in complex analysis and power series. Flashcards
What represents an infinite series of functions?
Define pointwise convergence.
Define uniform convergence.
What is the uniform limit theorem?
Prove the following theorem.
What is the Lemma about a test for uniform convergence?
Prove the following Lemma.
What is the Weierstrass M-test theorem?
What is the theorem about uniform convergence interchanging with integration?
Define locally uniform convergence.
What is the theorem about locally uniform convergence preserving continuity?
Define compact convergence.
What is the Lemma about compactness?
What is the proposition about compact convergence vs. locally uniform convergence?
What is the local Weierstress M-test theorem?
What is a power series?
What is the radius of convergence theorem?
What two tests are used to find the radius of convergence?
Ratio and root test
What is the locally uniform convergence of power series in the disk of convergence theorem?
Prove the following theorem.
Finish the following proposition.
What is Taylor’s theorem?
Prove the following theorem.
Prove the what the sum of a geometric series is.
