# The significance of Abel's Theorem

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?

• He's talking about some other, non-convergent series. – saulspatz Mar 3 at 15:55
• @SangchulLee That makes sense. Would you please convert your comment into an answer and let me accept it? – William McGonagall Mar 3 at 16:40
• No problem, and glad it helped :) – Sangchul Lee Mar 3 at 16:45

$${}^{\mathfrak{A}}\sum_{n=0}^{\infty} a_n := \lim_{x \uparrow 1} \sum_{n=0}^{\infty} a_n x^n$$
$${}^{\mathfrak{A}}\sum_{n=0}^{\infty}(-1)^n = \frac{1}{2}, \qquad {}^{\mathfrak{A}}\sum_{n=0}^{\infty}(-1)^n n^2 = 0, \qquad \text{etc.}$$