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).
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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Only do d, e and f
Transcribed Image Text:Let f(r) => anx" converges on an interval (-R, R) and let r, be a non zero sequence in
n=0
(-R, R) that converges to 0. If f(rn) = 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(rn) = 0 assume
n=0
that f(rn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R).
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