Prove that if a function has a power series expansion, its coefficients must match the Taylor series coefficients.

Uniqueness of Power Series Expansions

Series of numbers core

If \(f(x) = \sum_{n=0}^{\infty} a_n (x-a)^n\) for \(|x-a|<r\), prove that \(a_k = \frac{f^{(k)}(a)}{k!}\). In other words, if \(f(x)\) has a power series expansion about \(a\), that power series must be the Taylor series for \(f(x)\) about \(a\).

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What you need first

Power series
Differentiation of polynomials
Factorials

What this problem tests

Tests understanding of power series manipulation, specifically repeated differentiation and evaluation at the center.

This is a fundamental theorem in calculus and real analysis, establishing that power series representations are unique.