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

.
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To do:
The LHS needs to converge to as is chosen in the right way.

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