Can a power series be differentiated term by term at the end point provided the power series converges at one of its endpoints and the derived series also converges at that end point.And if a power series is not convergent at one of its endpoints,then can the derived series be convergent at that point.If not,then why? Can we prove it using only knowledge of differentiation,I was thinking something like domain of derived function is a subset of the domain of function.Is it correct?
Can I use the fact that $f$ is a function differentiable on $(a,b)$ and $f $ is defined on $[a,b]$ such that $\lim_{x\to b} f'(x)$ exists then $f$ is differentiable at $b$ and $f'(b)=\lim_{x\to b} f'(x)$.By the way is this statement true if $f$ is not continuous at $b$.I think true because We can use L Hospital's rule here.
Now if the derived power series were convergent at the end $R$ say.Now extend $f$ to $(-R,R]$ by assigning any value at $R$,not necessarily a continuous extension.Now $f'$ has limit at $R$ by Abel's theorem.So by the above theorem $f$ is differentiable at $R$ and $f'(R)=\lim_{x\to R} f'(x)=\sum{na_nR^{n-1}}$.Since,f is differentiable at $R$, $f$ is continuous at $R$.But we had assigned an arbitrary value at $R$.So $f$ need not be continuous at $R$,thus we have a contradiction.So,the derived series is not convergent at $R$.