Abel summation

Remarks

Example: for $a_n = (-1)^n (n + 1)$, that is $1 - 2 + 3 - 4 + \cdots$, one has $f(x) = \sum_{n \ge 0} (-1)^n (n+1) x^n = \frac{1}{(1 + x)^2}$ for $0 \le x < 1$. This tends to $\frac14$ as $x \to 1^-$, so the series is Abel summable to $1/4$.

Non-example: for $1 + 1 + 1 + \cdots$, condition 1 holds with $f(x) = \frac{1}{1 - x}$. But $f(x) \to +\infty$ as $x \to 1^-$, so condition 2 fails and the series is not Abel summable.

If $\sum a_n$ converges to $s$, then it is Abel summable to $s$; this is Abel's theorem on power series. In the notation of summability, one writes $A\text{-}\sum a_n = s$.