I should add a more focused remark.
Rereading K.Knopp's chapter about divergent summation it is better to relate it to the Riesz-method using generalized means (intead of referring to it as "modifed Euler-summation")
With the "summation of k'th order" k, the j'th terms of B (the partial sums) are here weighted with weights w_j such that
\( \lim_{n->\infty} s_n = \frac{w_0*b_0 + w_1*b_1 + w_2*b_2 + ....w_n*b_n}{w_0+w_1+w_2+...w_n} \)
where \( w_j = \frac{binomial(n,j)}{j!^k} \) (whether this makes sense or not...
)
I hope, this display makes it more clear.
Gottfried
Rereading K.Knopp's chapter about divergent summation it is better to relate it to the Riesz-method using generalized means (intead of referring to it as "modifed Euler-summation")
With the "summation of k'th order" k, the j'th terms of B (the partial sums) are here weighted with weights w_j such that
\( \lim_{n->\infty} s_n = \frac{w_0*b_0 + w_1*b_1 + w_2*b_2 + ....w_n*b_n}{w_0+w_1+w_2+...w_n} \)
where \( w_j = \frac{binomial(n,j)}{j!^k} \) (whether this makes sense or not...
)I hope, this display makes it more clear.
Gottfried
Gottfried Helms, Kassel