Abel's Theorem and Uniform Convergence

Abel's theorem addresses the subtle question of convergence at the boundary of the interval of convergence, while the Weierstrass M-test provides a systematic criterion for uniform convergence of series.

Abel's Theorem

Theorem7.6Abel's Theorem

If the power series $f(x) = \sum_{n=0}^\infty c_n x^n$ has radius of convergence $R$ and the series $\sum_{n=0}^\infty c_n R^n$ converges to a finite value $S$ , then $\lim_{x \to R^-} f(x) = S = \sum_{n=0}^\infty c_n R^n$ That is, if the power series converges at an endpoint, the function defined by the series is continuous from the interior at that endpoint.

ExampleApplication: $\ln 2$

The series $\ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} x^n$ has $R = 1$ . At $x = 1$ , the alternating harmonic series $\sum (-1)^{n-1}/n$ converges (by the alternating series test). Abel's theorem gives $\ln 2 = \lim_{x \to 1^-} \ln(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$

Uniform Convergence

Theorem7.7Weierstrass M-Test

Let $\{f_n\}$ be a sequence of functions defined on a set $S$ . If there exist constants $M_n \geq 0$ such that $|f_n(x)| \leq M_n$ for all $x \in S$ and $\sum M_n < \infty$ , then $\sum f_n$ converges uniformly and absolutely on $S$ .

ExampleUniform convergence of power series

The series $\sum_{n=0}^\infty x^n / n!$ converges uniformly on $[-A, A]$ for any $A > 0$ : take $M_n = A^n/n!$ , then $\sum M_n = e^A < \infty$ . However, the convergence is not uniform on all of $\mathbb{R}$ since the partial sums require more terms for larger $|x|$ .

Consequences of Uniform Convergence

RemarkWhy uniform convergence matters

Uniform convergence of a series $\sum f_n(x)$ ensures:

If each $f_n$ is continuous, then $\sum f_n$ is continuous
Term-by-term integration: $\int \sum f_n = \sum \int f_n$
Term-by-term differentiation (under additional hypotheses): $(\sum f_n)' = \sum f_n'$

Power series converge uniformly on any closed subinterval of their interval of convergence, which is why the operations of Theorem 7.3 (term-by-term differentiation and integration) are valid. The failure of uniform convergence at boundary points is precisely what makes endpoint analysis delicate.