power series convergence
i was reading a proof on :
given that the power series converges on (-R,R) then it is continuous on the open interval(-R,R).
i know that i have to split it up into using triangular inequality where for all ε>0
δ'>o st l f(x) - f_m(x) l < ε/3
δ'' >0 st l f_m(x) - f_m(c) l < ε/3 ( continous at c).
im wondering, does δ' and δ'' depend on ε or x?
This is a relatively difficult theorem. Its proof pretty much goes along the lines of the proof of Abel's Theorem.
It's not true for series of functions in general; for example, trigonometric series may be discontinuous. This is something special about power series.
does δ depend on ε in the proof?
It very well can, yes, as for any proof that a function is continuous at a point.