So I get to Abel's Theorem and I'm like, "What the fork? I think I've seen this before."
THEOREM III. Suppose that the series ∑anxn has a positive or infinte radius of convergence R. Let 0 < r < R. Then the series converges uniformly on the closed interval -r ≤ x ≤ r.
THEOREM XI. If ∑anxn converges at x = R, then it converges uniformly on the closed interval 0 ≤ x ≤ R. A like conclusion
golds for -R ≤ x ≤ 0 if the series converges at x = -R.
The Attempt at a Solution
Theorem 11 seems to be restating theorem 3 in a different way.
Theorem 2 mentions the "radius of convergence R." Theorem 11 mentions an R at which the series converges, which means R is any number inside the radius of convergence. Thus we're talking about the interval of convergence in both theorems. Then, basically, they both state that a series converges uniformly in its interval of convergence.
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