Given a Taylor series
 |
(1)
|
where the complex number 
has been written in the polar form 
, examine the real
and imaginary parts
 |
(2)
|
 |
(3)
|
Abel's theorem states that, if 
and 
are convergent, then
 |
(4)
|
Stated in words, Abel's theorem guarantees that, if a real power series converges for some positive value of the argument, the domain of uniform convergence extends at least up to and including this point. Furthermore, the continuity of the sum function extends at least up to and including this point.
