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I am having hard time understanding Abel's Theorem as well as where exactly it is used, though I have some crude idea that it is useful in Analysis.

Here are a few sources I checked.

I should say, I was not able to grasp either of those definitions perfectly, though wikipedia was far better.

I would appreciate it if someone could give me a more intuitive explanation of Abel's Theorem with Use in Context.

Also, why is limit in both cases taken as $\lim_{ x \rightarrow 1^-}$ and not $\lim_{ x \rightarrow R^-}$ where $R$ is radius of convergence.

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Regarding $\to1^-$ vs. $\to R^-$, see the section "Remark" of Wikipedia. The $R$ "hides" in the $z$. – Hagen von Eitzen Dec 30 '12 at 22:31
The Applications section of the Wikipedia article has examples of how the theorem is used. – Robert Israel Dec 30 '12 at 22:34
What is the question? – Did Dec 30 '12 at 22:39
@HagenvonEitzen, because I was not expecting the answer to be there, I completely missed that, but now I got it. – 007resu Dec 30 '12 at 22:45
@did, second last line. – 007resu Dec 30 '12 at 22:46
up vote 2 down vote accepted

Let $\{a_k\}_{k=1}^{\infty}$ be a sequence of complex numbers such that $\sum_{k=1}^{\infty} z^k a_k$ exists for all $|z| < 1$. We say that the series is Abel summable if $ \displaystyle \lim_{|z| \to 1^-}\sum_{k=1}^{\infty} z^k a_k$ exists. The idea is this. It may not actually be the case that $\sum_{k=1}^{\infty} a_k$ exists, or, more generally, we may not be certain that it exists. The claim that $\lim _{|z| \to 1^-}\sum_{k=1}^{\infty} z^k a_k$ exists is a strictly weaker statement (as can be seen by applying the DCT). So we might be able to prove that summability holds first by considering Abel sums.

A major utility of alternative forms of summation - such as Cessaro or Abel summation - occurs by extending properties of summable series to a wider class. Consider the following example: let $f:\mathbb{T} \to \mathbb{R}$ be an integrable function. It is not, in general, true that if $a_k$ is the $k$th Fourier coefficient of $f$ that $f(x) = \sum_{k \in \mathbb{Z}} a_k e^{ikx}$, or that the right hand side even converges. However, it is somewhat straightforward to show that the series Abel sums to $f$ wherever $f$ is continuous. This can in turn be used to solve the steady state heat equation on the unit disc.

It also has the immediate corollary that if $f$ and $g$ are $2\pi$ periodic and have identical Fourier coefficients, then $f=g$ at points of continuity.

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