(f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that f(rn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R). %3D
(f) Assume that g(x) > bna" is convergent on (-R, R). Instead of assuming that f(rn) = 0 assume that f(rn) = g(xn) for all n. Show that f(x) = g(x) on (-R, R). %3D
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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do F only
Transcribed Image Text:D2: Let f(x) = 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)
> bna" is convergent on (-R, R). Instead of assuming that
n=0
f(In) = 0 assume that f(xn) = g(n) for all n. Show that f(x) = g(x) on (–R, R).
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