The following theorem is in Rudin's Principles of Mathematical Analysis, 8.2:
Suppose $c_n \geq 0$ and $S = \sum c_n$ converges. Put $f(x) = \sum_{n=0}^{\infty} c_nx^n$. Show that
$$
\lim_{x \rightarrow 1} f(x) = \sum_{n=0}^{\infty} c_n.
$$
Rudin goes on to give a somewhat involved proof using Abel Summation. I'm wondering why this is necessary; why not just this argument?
"Proof:" Pick $N$ such that $\sum_{n > N} c_n < \epsilon$. Pick $r$ such that $1-r^N < \epsilon/S$. Then we have
$$
\sum_{n=0}^{\infty} c_n – \sum_{n=0}^{\infty} c_nr^n = \sum_{n=0}^N c_n(1-r^n) + \sum_{n>N} c_n(1-r^n) < 2 \epsilon.
$$
Briefly, Rudin's method involves using Abel summation and then bounding the new sequence of partial sums, which seems like a less natural approach to me. Does anyone have a reason why he did that?
Best Answer
I agree heartily with Qiaochu: the case in which all terms are non-negative is a trivial case of Abel's Theorem. This comes out in my take on Abel's Theorem in these notes (Wayback Machine) on an undergraduate analysis class I taught a few years ago.
Note that I begin by giving exactly the proof from Rudin (why not?). After that I have some discussion of "Abel summation" and applications of this result. Note in particular Exercise 40: a series with non-negative terms is Abel-summable iff it is convergent in the usual sense.
It might also help to see some typical applications of Abel's Theorem, which are to non-convergent series. In my notes, I first use it to prove that if the Cauchy product of two convergent series converges at all, then it converges to the expected value, namely the product of the two sums. Note that earlier on in my notes I had proved that when at least one of the two series is absolutely convergent the Cauchy product necessarily converges to the expected value, whereas there are examples of divergent Cauchy products of two nonabsolutely convergent series. Then I use Abel's Theorem to justify the "calculus identity"
$\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}$.
In both cases the setting is of non-absolute convergence.
I haven't looked back at my copy of Rudin, but I am willing to believe that he does not explicitly mention the significance of the non-absolute convergence here. If so, this is an instance of his book being a little too laconic for contemporary eyes.
Added: Okay, I just looked back at Rudin's book. First, he also follows the statement and proof of the theorem with the application to Cauchy products (what a coincidence...). Second -- it's true: he really should hit a little harder the point that the content comes when the terms are not all non-negative, and moreover when the series is not absolutely convergent. (Nevertheless Chapter 8 is by far my favorite in Rudin's book: a lot of classical content is covered in a very clean, efficient way.)