Sequences and Series/Power series

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