Basic Analysis I — Real Analysis · Problem 6.2.20

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · power-series-identity-theorem

Problem

(requires \S2.6 and \S4.3): Suppose f(x):=∑n=0∞cn(x−a)nf(x) \mathrel{:=} \sum_{n=0}^\infty c_n(x-a)^n converges in (a−ρ,a+ρ)(a - \rho, a + \rho). (a)) Suppose that f(k)(a)=0f^{(k)}(a) = 0 for all k=0,1,2,3,…k = 0, 1, 2, 3, \ldots. Prove that cn=0c_n = 0 for all nn, or in other words, f(x)=0f(x) = 0 for all x∈(a−ρ,a+ρ)x \in (a - \rho, a + \rho). (b)) Using part a) prove a version of the so-called ``identity theorem for analytic functions'': If there exists an ϵ>0\epsilon > 0 such that f(x)=0f(x) = 0 for all x∈(a−ϵ,a+ϵ)x \in (a - \epsilon, a + \epsilon), then f(x)=0f(x) = 0 for all x∈(a−ρ,a+ρ)x \in (a - \rho, a + \rho).

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