Lebl-Basic-Analysis-I · Problem 6.2.20

Sequences of Functions

Mathematics · power-series-identity-theorem

Problem

(requires §2.6 and §4.3) 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).

Related theory

Browse the problem library