## Corollary 8.3.11: Power Series is infinitely often Differentiable

If a power series

*f(x) = a*has radius of convergence_{n}(x-c)^{n}*r*, then*f*is infinitely often differentiable for*|x-c| < r*. In other words,*f C(c-r, c+r)*.By our previous theorem a power series with radius of convergence

r = lim sup |a_{n}|/|a_{n+1|}

can be differentiated term by term, so that

f'(x) = n a_{n}(x-c)^{n-1}

But the center of this power series is again *c* and the
radius of convergence is also

lim sup |n a_{n}|/|(n+1) a_{n+1|}= lim sup n/(n+1) |a_{n}|/|a_{n+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 ...