On p.415 of Bartle's The Elements of Real Analysis, Abel's Theorem is stated as follows:
28.20 ABEL'S THEOREM. Suppose that the power series $\sum_{n=0}^\infty(a_nx^n)$ converges to $f(x)$ for $|x|<1$ and that $\sum_{n=0}^\infty(a_n)$ converges to $A$. Then the power series converges uniformly in $I=[0,1]$ and $$\lim_{x\to1-}f(x)=A.$$
On next page, Bartle wrote (emphasis mine):
One of the most interesting things about this result is that it suggest (sic.) a method of attaching a limit to series which may not be convergent. Thus, if $\sum_{n=1}^\infty(b_n)$ is an infinite series, we can form the corresponding power series $\sum_{n=1}^\infty(b_nx^n)$. If the $b_n$ do not increase too rapidly, this power series converges to a function $B(x)$ for $|x|<1$. If $B(x)\to\beta$ as $x\to1-$, we say that the series $\sum(b_n)$ is Abel summable to $\beta$…. The content of Abel's Theorem 28.20… asserts that if a series is already convergent, then it is Abel summable to the same limit.
Since both $\sum_{n=1}^\infty(a_nx^n)$ (with $|x|<1$) and $\sum_{n=1}^\infty(a_n)$ are assumed to be convergent in the theorem, can somebody please explain what is "series which may not be convergent" in Bartle's discussion?
Best Answer
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He is simply telling that the theorem provides an inspiration for defining new type of infinite summation, called Abel summation. It is defined as
$$ {}^{\mathfrak{A}}\sum_{n=0}^{\infty} a_n := \lim_{x \uparrow 1} \sum_{n=0}^{\infty} a_n x^n $$
(although the notation varies from literature to literature), and in such case, Abel's theorem asserts that 'Abel summability generalizes ordinary summability'. But this also assigns value to series which is not convergent in ordinary summbability, such as
$$ {}^{\mathfrak{A}}\sum_{n=0}^{\infty}(-1)^n = \frac{1}{2}, \qquad {}^{\mathfrak{A}}\sum_{n=0}^{\infty}(-1)^n n^2 = 0, \qquad \text{etc.} $$