Corollary 8.3.11: Power Series is infinitely often Differentiable
By our previous theorem a power series with radius of convergence
r = lim sup |an|/|an+1|
can be differentiated term by term, so that
f'(x) = n an (x-c)n-1
But the center of this power series is again c and the radius of convergence is also
lim sup |n an|/|(n+1) an+1| = lim sup n/(n+1) |an|/|an+1| = r
Thus, the derivative itself is a power series with center c and radius of convergence r. Thus, by the same theorem, it can be differentiate again ... and again ... and again ...