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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 = ∞ ....
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