This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MAT A26 Lecture 45 1 Power Series definition 1.1. A power series is a series of the form ∞ X k =0 a k ( z a ) k , where a k is a sequence and a is a constant. The power series can be a real or complex depending on whether a k and a are real or complex. corollary 1.2. Convergence for Power Series Let ∑ ∞ k =0 a k ( z a ) k be a given power series, then there exists a unique number R ∈ [0 , ∞ ) , called the radius of convergence such that 1. the series converges absolutely for  z a  < R 2. the series diverges for  z a  > R 3. the test fails if  z a  = R proof. The proof of this theorem is provided in the course Complex Vari ables (MATC34H3) So stay in suspense until you take that course. It is worth the wait!! Summary: There are three cases: Case 1: ∑ ∞ k =0 a k ( z a ) k converges absolutely for all z ∈ C , in which case we say that the radius of convergence R = ∞ ....
View
Full
Document
This note was uploaded on 03/04/2012 for the course MATH 10250 taught by Professor Himonas during the Fall '08 term at Notre Dame.
 Fall '08
 HIMONAS
 Calculus, Power Series

Click to edit the document details