8.3. Series and Power Series

Corollary 8.3.11: Power Series is infinitely often Differentiable

If a power series f(x) = an (x-c)n has radius of convergence 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 |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 ...

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