Power Series

analysis

Definition

Power Series

A power series is a series of form:

where is a sequence, being called the expansion point of and are called the coefficients of that power series.

With being the convergence radius of the power series, then is continuous within the convergence radius .

Differentiability

Let a power series with convergence radius . Then is differentiable in that area. The derivative is given by term-wise differentiation, i.e., for all with it holds:

The convergence radius of the derived series is also .

Uniqueness Theorem

Definition

Uniqueness Theorem for Power Series

Suppose the function admits a power series representation in some epsilon neighbourhood of , i.e.

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

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