Explain the difference between these two theorems.

[Jamin2112]

Homework Statement

So I get to Abel's Theorem and I'm like, "What the fork? I think I've seen this before."

Homework Equations

THEOREM III. Suppose that the series ∑a_nxⁿ has a positive or infinite radius of convergence R. Let 0 < r < R. Then the series converges uniformly on the closed interval -r ≤ x ≤ r.

THEOREM XI. If ∑a_nxⁿ 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. [PLAIN]http://images2.memegenerator.net/ImageMacro/4879227/lolwut.jpg?imageSize=Large&generatorName=OWL

 

[CompuChip]

The first one says, that if it is has radius of convergence R, then it converges uniformly for all points within that radius (but it says nothing about the case where |x| = R).

According to the second one, if it converges at x = R, then the radius of convergence is at least R (and then it reminds you that this implies that it also converges for all |x| < R).

Have to admit I had to look twice to notice the difference though, so I understand your confusion.

 

[micromass]

Also note that the first theorem does NOT say that the series converges uniformly on it's interval of convergence. It says that the series converges uniformly on something smaller than the interval of convergence.

For example, if the interval of convergence is [-2,2]. Then the first theorem can be applied to show that convergence is uniform on [-1,1] or [-1.5, 1.5], but NOT on [-2,2]!

If you want uniform convergence on the full interval of convergence, then you need to apply the second theorem.

 

[Jamin2112]

Thanks, guys! So Theorem 11 is more powerful since it says something about the point x = R and uniform convergence?

 

[micromass]

Yes, theorem 11 has a more powerful conclusion!

 

[CompuChip]

Do note that the first theorem has a weaker assumption, too.
It is of the form "if A, then B for x < R"
The second on is of the form "if A and B for x = R, then C for x < R as well as for x = R".