
# 8: Back to Power Series

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• 8.1: Uniform Convergence
We will now draw our attention back to the question that originally motivated these deﬁnitions, “Why are Taylor series well behaved, but Fourier series are not necessarily?” More precisely, we mentioned that whenever a power series converges then whatever it converged to was continuous. Moreover, if we diﬀerentiate or integrate these series term by term then the resulting series will converge to the derivative or integral of the original series. This was not always the case for Fourier series.
• 8.2: Uniform Convergence: Integrals and Derivatives
We saw in the previous section that if f(n) is a sequence of continuous functions which converges uniformly to f on an interval, then f must be continuous on the interval as well. This was not necessarily true if the convergence was only pointwise, as we saw a sequence of continuous functions deﬁned on (−∞,∞) converging pointwise to a Fourier series that was not continuous on the real line. Uniform convergence guarantees some other nice properties as well.
• 8.3: Radius of Convergence of a Power Series
We’ve developed enough machinery to look at the convergence of power series.
• 8.4: Boundary Issues and Abel’s Theorem
The integrations we performed in Chapter 2 are legitimate due to the Abel's theorem which extends uniform convergence to the endpoints of the interval of convergence even if the convergence at an endpoint is only conditional. Abel did not use the term uniform convergence, as it hadn’t been deﬁned yet, but the ideas involved are his.

Thumbnail: Niels Henrik Abel. Image used with permission (public domain).

### Contributors

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)