Power Series

analysis

Definition

Power Series

A power series is a series of form:

$$f(x)=\sum _{n\ge 0}{a}_n(x-x_0{)}^n$$

where $(a_n{)}_{n\in \mathbb{N}}$ is a sequence, $x_0$ being called the expansion point of $f(x)$ and $a_n$ are called the coefficients of that power series.

With $R$ being the convergence radius of the power series, then $f$ is continuous within the convergence radius $|x-x_0|<R$.

Differentiability

Let $f(x)={\sum }_{n\ge 0}{a}_n(x-x_0{)}^n$ a power series with convergence radius $R$. Then $f(x)$ is differentiable in that area. The derivative is given by term-wise differentiation, i.e., for all $x$ with $|x-x_0|<R$ it holds:

$$f^{\prime }(x)=\sum _{n\ge 1}n{a}_n(x-x_0{)}^{n-1}$$

The convergence radius of the derived series is also $R$.

Uniqueness Theorem

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.

Link to original

]