Let f(r) = > anx" converges on an interval (-R, R) and let rn be a non zero sequence in n=0 (-R, R) that converges to 0. If f(r,n) = 0 for every n show that (a) f(0) = 0 (b) f'(0) = 0 (c) There exists another non zero sequence in (-R, R), Yn, that converges to 0, such that f'(yn) = 0 for every n. (d) Show that f"(0) = 0. (e) Use induction and the ideas from the above to show that f(k)(0) = 0 for every k and conclude that f = 0. (f) Assume that g(x) = > ana" on (-R, R). Instead of assuming that f(xn) = 0 assume n=0 that f(xn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R).

Let f(r) = > anx" converges on an interval (-R, R) and let rn be a non zero sequence in n=0 (-R, R) that converges to 0. If f(r,n) = 0 for every n show that (a) f(0) = 0 (b) f'(0) = 0 (c) There exists another non zero sequence in (-R, R), Yn, that converges to 0, such that f'(yn) = 0 for every n. (d) Show that f"(0) = 0. (e) Use induction and the ideas from the above to show that f(k)(0) = 0 for every k and conclude that f = 0. (f) Assume that g(x) = > ana" on (-R, R). Instead of assuming that f(xn) = 0 assume n=0 that f(xn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 72E

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