# Sequences and Series/Power series

**Proposition (identity theorem for one-dimensional power series)**:

Let

- and

be two (complex or real) power series that converge on for some . Suppose that is an accumulation point of the set . Then we have for all .

**Proof:** Assume that not for all . Then there exists a least (call it ) such that . Consider the function

- ,

which is defined on at least . Since for , the power series starts at . Therefore,

is a well-defined function on which is also continuous due to the continuity of power series. Moreover,

- ,

and by continuity of , there exists a such that for all . But by definition,

- ,

so that we have for that and consequently , and hence . But this contradicts the assumption that was an accumulation point of .

**Example (falsity of the identity theorem for multi-dimensional power series)**:

For multi-dimensional power series, that is power series of the type

- for a ,

the set may have as an accumulation point even when does not vanish. An easy example (which works in any dimension ) is and

- .

**Theorem (Abel's theorem)**:

Let

be a real or complex power series of convergence radius , and suppose that

- .

Then

- .

{{proof|By Abelian partial summation, we have

for and , where we denote as usual

- .

Substituting , we get

- .

We then put