TITLE: Reference request: Behavior of power series at endpoints
QUESTION [7 upvotes]: I would like to find a calculus book (or a book on real analysis or advanced calculus) which has the following result:
If a power series $\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n}$ has a radius of convergence $r>0$, and the power series diverges at one of the endpoints, then
$\displaystyle\sum_{n=1}^{\infty}n a_{n}x^{n-1}$ also diverges at that endpoint.
(I'm not asking for a proof, just a reference.)

REPLY [2 votes]: You can find this, a proof of it, and an in depth exploration of the behavior of power series at their boundaries in E.C. Titchmarsh's "Theory of Functions."