Please solve as soon as possible 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 atevery point z satisfying |z – zol < |Z1 – zo|-Ro2. Define Ro := sup {|z – zo| : f(z) converges}.Then f(z) converges absolutely whenever |z – zo| < Ro and divergeswhenever |z – zo| > Ro.31. By the nth term test, the sequence (a,(z1 – zo)") converges (to 0) and is therefore boundedProof.by some M e R+. Thus|an| |z – zo|" = |an||21 – z0l" (-20)|z – zol<1< Mr" where r=%3D|21 - zolSince 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) convergesand |z – zol < |zı – zol: now apply part (a). The remaining part is an exercise.

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Answered: Consider a power series f(z) = Ean(z –…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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