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Consider a power series f(z) = Ean(z – zo)". 1. If f converges at a point z1 # z0, then it is absolutely convergent at every point z satisfying |z – zol < |Z1 – zo|- Theorem 5.8. Ro z1 zo 2. Define Ro := sup {|z – zo| : f(z) converges}. Then f(z) converges absolutely whenever |z – zo < Ro and diverges whenever |z – zo| > Ro-

Consider a power series f(z) = Ean(z – zo)". 1. If f converges at a point z1 # z0, then it is absolutely convergent at every point z satisfying |z – zol < |Z1 – zo|- Theorem 5.8. Ro z1 zo 2. Define Ro := sup {|z – zo| : f(z) converges}. Then f(z) converges absolutely whenever |z – zo < Ro and diverges whenever |z – zo| > Ro-

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Theorem 5.8. Consider a power series f(z) = Ean(z – zo)". 1. If f converges at a point z1 # z0, then it is absolutely convergent at every point z satisfying |z – zol < |Z1 – zo|- Ro 2. Define Ro := sup {|z – zo| : f(z) converges}. Then f(z) converges absolutely whenever |z – zo| < Ro and diverges whenever |z – zo| > Ro. 3 1. By the nth term test, the sequence (a,(z1 – zo)") converges (to 0) and is therefore bounded Proof. by some M e R+. Thus |an| |z – zo|" = |an||21 – z0l" (-20) |z – zol <1 < Mr" where r= %3D |21 - zol Since E Mr" converges, we conclude (comparison test) that E la,||2 – zo|" converges. 2. By standard properties of the supremum, if |z – zo| < Ro, then 3z1 such that f(z1) converges and |z – zol < |zı – zol: now apply part (a). The remaining part is an exercise.
Transcribed Image Text:Theorem 5.8. Consider a power series f(z) = Ean(z – zo)". 1. If f converges at a point z1 # z0, then it is absolutely convergent at every point z satisfying |z – zol < |Z1 – zo|- Ro 2. Define Ro := sup {|z – zo| : f(z) converges}. Then f(z) converges absolutely whenever |z – zo| < Ro and diverges whenever |z – zo| > Ro. 3 1. By the nth term test, the sequence (a,(z1 – zo)") converges (to 0) and is therefore bounded Proof. by some M e R+. Thus |an| |z – zo|" = |an||21 – z0l" (-20) |z – zol <1 < Mr" where r= %3D |21 - zol Since E Mr" converges, we conclude (comparison test) that E la,||2 – zo|" converges. 2. By standard properties of the supremum, if |z – zo| < Ro, then 3z1 such that f(z1) converges and |z – zol < |zı – zol: now apply part (a). The remaining part is an exercise.
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