It is said that:
Abel summation and Euler summation are not comparable.
We were able to find examples of divergent series which are Euler summable but not Abel summable, for instance $$ 1-2+4-8+16-\dots$$
However, we couldn't find any example of a divergent series which is Abel summable but not Euler summable.
Do you know such an example?
Thank you!
EDIT: Dear Peter, this is the definition of Euler summation:
Let $\sum_{n=0}^\infty a_n$ be any series. The Euler transformation of this series is defined as: \begin{equation*} \sum_{n=0}^\infty \frac{1}{2^{n+1}}b_n\quad\text{ with }\quad b_n:=\sum_{k=0}^n\left(\begin{array}[h]{c} n \\ k \end{array}\right) a_k \end{equation*}
The series $\sum_{n=0}^\infty a_n$ is called Euler summable if the Euler transformation of this series $$\sum_{n=0}^\infty \frac{1}{2^{n+1}}b_n$$ is converges in the usual sense.
The Euler sum is then given by $$\sum_{n=0}^\infty \frac{1}{2^{n+1}}b_n.$$