Uniqueness Theorem for Power Series

analysis

Definition

Uniqueness Theorem for Power Series

Suppose the function $f(x)$ admits a power series representation in some epsilon neighbourhood of $x_0$, i.e.

$$\forall x\in U_{\epsilon }(x_0):f(x)=\sum _{n=0}^{\infty }{a}_n(x-x_0{)}^n$$

Then this representation is unique. In other words, if a function admits a power series expansion, that expansion is uniquely determined.