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I have a little question concerning abel-summmation.

In some books the n-th abel-mean of a sequence $(x_n) \subset \mathbb{K}$ is defined as: $$ A_{n,r}[x_n] = (1 - r) \sum_{k=0}^{n} x_k r^k $$

In other books the abel-means are defined as: $$ A_{n,r}[x_n] = \sum_{k=0}^{n} x_k r^k $$

In both sources it says, that $(x_n)$ converges to $x$ is the sense of abel if $$ \lim_{r \nearrow 1}\lim_{n\to\infty}A_{n,r}[x_n] = x $$

Here is my question: Are these two concepts equivalent? Are the limits (if exist) in both cases the same? With best regards, Mat

Mat
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The first version is a mean, in the sense that, if $\mathbf x=(x_n)_n$ with $x_n=x$ for every $n$, then $A_{n,r}(\mathbf x)=(1-r^{n+1})x$ hence $A_{n,r}(\mathbf x)\to x$ when $n\to\infty$, for every $0\lt r\lt1$. Likewise, if $a\leqslant x_n\leqslant b$ for every $n$, then $$ \liminf\limits_{r\to1^-}\liminf\limits_{n\to\infty} A_{n,r}(\mathbf x)\geqslant a,\qquad \limsup\limits_{r\to1^-}\limsup\limits_{n\to\infty} A_{n,r}(\mathbf x)\leqslant b, $$ which is also a desirable property for a mean.

The second version is not a mean, for example, if $\mathbf x=(x_n)_n$ with $x_n=x$ for every $n$ and $x\ne0$, then $A_{n,r}(\mathbf x)\to x/(1-r)$ when $n\to\infty$ and the limit of that when $r\to1^-$ is infinity, not $x$.

Edit Let me suggest reading this previous answer due to @Byron Schmuland.

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