(C, α)/(C,k) summation

Question

I'm stuck here at a definition in Hardy's "Divergent Series". On page 96 he defines the Cesàro means $(C,k)$ by
We write
$A_n^{0} = A_n = a_0 + a_1 + ... + a_n \quad , ..., \quad A_n^k = A_0^{k-1} + A_1^{k-1} + ... + A_n^{k-1}$
and $E_n^k$ for the value of $A_n^k$ when $a_0 = 1$ and $a_n = 0$ for $n>0$ , i.e. , when $A_n = 1$ for all $n$. If
$ C_n^k(A) = \frac{A_n^k}{E_n^k} \to A$
when $n\to\infty$, then we say that $\sum a_n$ is summable $(C,k)$ to sum $A$, and write $a_0+a_1+a_2 + ... = A \quad (C,k).$
What I don't really understand here (maybe because of my English) is what happens with $E_n^k$ if $A_n$ is not 1 + 0 + 0 + 0 + ... It's written that $(C, k=1)$ is the usual Cesàro summation, that one however, can sum 1-1+1-1+1-1+... while not having to meet the conditions for the $(C,k)$ summation. What am I overlooking/misunderstanding? Thanks a lot!

Answer 1

This answer is 1:1 copied from my answer on mathoverflow to prevent loss, if MO closes your question there

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The question on $E^k_n$ shows a somehow misleaded intuition here, so I think it would be good to show an example display of the process.

If we assume the process of building partial sums, and partial sums of partial sums, and etc, of an initial sequence $A$ as an iterated operation D , then we compare the intermediate results of $D^k$ applied on $A$ with $D^k$ applied to $E$ where $E$ is defined as a constant $E=[1,0,0,...]$

The $k$-times application of operation $D$ on $A$ gives the sequence $A^k$ and on $E$ gives $E^k$ , and then we simply divide the values $A^k$ by $E^k$.
This is somehow a normalizing of the operation $D$ .

Here I show the partial results for $A^0=[1,-2,3,-4,5,-6,...]$ up to $k=4$ . We see, that for $k=3$ the values in the comlumn $A/E$ approach visibly the expected value $0.25$ and for $k=4$ it seems to be the same, only with slower convergence. So application of Cesarosum with order $3$ should give the best approximation with least terms involved while the results for $k<3$ are not converging.

$$ \Tiny \begin{array} {rrr|rrr|rrr|rrr|rrr} A^0& E^0& A/E & A^1& E^1& A/E & A^2& E^2& A/E & A^3& E^3& A/E & A^4& E^4& A/E & \\ \hline 1 & 1 & 1.00000 & 1 & 1 & 1.00000 & 1 & 1 & 1.00000 & 1 & 1 & 1.00000 & 1 & 1 & 1.00000 \\ -2 & 0 & 0 & -1 & 1 & -1.00000 & 0 & 2 & 0 & 1 & 3 & 0.333333 & 2 & 4 & 0.500000 \\ 3 & 0 & 0 & 2 & 1 & 2.00000 & 2 & 3 & 0.666667 & 3 & 6 & 0.500000 & 5 & 10 & 0.500000 \\ -4 & 0 & 0 & -2 & 1 & -2.00000 & 0 & 4 & 0 & 3 & 10 & 0.300000 & 8 & 20 & 0.400000 \\ 5 & 0 & 0 & 3 & 1 & 3.00000 & 3 & 5 & 0.600000 & 6 & 15 & 0.400000 & 14 & 35 & 0.400000 \\ -6 & 0 & 0 & -3 & 1 & -3.00000 & 0 & 6 & 0 & 6 & 21 & 0.285714 & 20 & 56 & 0.357143 \\ 7 & 0 & 0 & 4 & 1 & 4.00000 & 4 & 7 & 0.571429 & 10 & 28 & 0.357143 & 30 & 84 & 0.357143 \\ -8 & 0 & 0 & -4 & 1 & -4.00000 & 0 & 8 & 0 & 10 & 36 & 0.277778 & 40 & 120 & 0.333333 \\ 9 & 0 & 0 & 5 & 1 & 5.00000 & 5 & 9 & 0.555556 & 15 & 45 & 0.333333 & 55 & 165 & 0.333333 \\ -10 & 0 & 0 & -5 & 1 & -5.00000 & 0 & 10 & 0 & 15 & 55 & 0.272727 & 70 & 220 & 0.318182 \\ 11 & 0 & 0 & 6 & 1 & 6.00000 & 6 & 11 & 0.545455 & 21 & 66 & 0.318182 & 91 & 286 & 0.318182 \\ -12 & 0 & 0 & -6 & 1 & -6.00000 & 0 & 12 & 0 & 21 & 78 & 0.269231 & 112 & 364 & 0.307692 \\ 13 & 0 & 0 & 7 & 1 & 7.00000 & 7 & 13 & 0.538462 & 28 & 91 & 0.307692 & 140 & 455 & 0.307692 \\ -14 & 0 & 0 & -7 & 1 & -7.00000 & 0 & 14 & 0 & 28 & 105 & 0.266667 & 168 & 560 & 0.300000 \\ 15 & 0 & 0 & 8 & 1 & 8.00000 & 8 & 15 & 0.533333 & 36 & 120 & 0.300000 & 204 & 680 & 0.300000 \\ -16 & 0 & 0 & -8 & 1 & -8.00000 & 0 & 16 & 0 & 36 & 136 & 0.264706 & 240 & 816 & 0.294118 \\ 17 & 0 & 0 & 9 & 1 & 9.00000 & 9 & 17 & 0.529412 & 45 & 153 & 0.294118 & 285 & 969 & 0.294118 \\ -18 & 0 & 0 & -9 & 1 & -9.00000 & 0 & 18 & 0 & 45 & 171 & 0.263158 & 330 & 1140 & 0.289474 \\ 19 & 0 & 0 & 10 & 1 & 10.0000 & 10 & 19 & 0.526316 & 55 & 190 & 0.289474 & 385 & 1330 & 0.289474 \\ -20 & 0 & 0 & -10 & 1 & -10.0000 & 0 & 20 & 0 & 55 & 210 & 0.261905 & 440 & 1540 & 0.285714 \\ 21 & 0 & 0 & 11 & 1 & 11.0000 & 11 & 21 & 0.523810 & 66 & 231 & 0.285714 & 506 & 1771 & 0.285714 \\ -22 & 0 & 0 & -11 & 1 & -11.0000 & 0 & 22 & 0 & 66 & 253 & 0.260870 & 572 & 2024 & 0.282609 \\ 23 & 0 & 0 & 12 & 1 & 12.0000 & 12 & 23 & 0.521739 & 78 & 276 & 0.282609 & 650 & 2300 & 0.282609 \\ -24 & 0 & 0 & -12 & 1 & -12.0000 & 0 & 24 & 0 & 78 & 300 & 0.260000 & 728 & 2600 & 0.280000 \\ 25 & 0 & 0 & 13 & 1 & 13.0000 & 13 & 25 & 0.520000 & 91 & 325 & 0.280000 & 819 & 2925 & 0.280000 \\ -26 & 0 & 0 & -13 & 1 & -13.0000 & 0 & 26 & 0 & 91 & 351 & 0.259259 & 910 & 3276 & 0.277778 \\ 27 & 0 & 0 & 14 & 1 & 14.0000 & 14 & 27 & 0.518519 & 105 & 378 & 0.277778 & 1015 & 3654 & 0.277778 \\ -28 & 0 & 0 & -14 & 1 & -14.0000 & 0 & 28 & 0 & 105 & 406 & 0.258621 & 1120 & 4060 & 0.275862 \\ 29 & 0 & 0 & 15 & 1 & 15.0000 & 15 & 29 & 0.517241 & 120 & 435 & 0.275862 & 1240 & 4495 & 0.275862 \\ -30 & 0 & 0 & -15 & 1 & -15.0000 & 0 & 30 & 0 & 120 & 465 & 0.258065 & 1360 & 4960 & 0.274194 \\ 31 & 0 & 0 & 16 & 1 & 16.0000 & 16 & 31 & 0.516129 & 136 & 496 & 0.274194 & 1496 & 5456 & 0.274194 \\ -32 & 0 & 0 & -16 & 1 & -16.0000 & 0 & 32 & 0 & 136 & 528 & 0.257576 & 1632 & 5984 & 0.272727 \\ \vdots& \vdots&\vdots & \vdots& \vdots& \vdots & \vdots& \vdots& \vdots & \vdots& \vdots& \vdots &\vdots& \vdots& \vdots & \\ \end{array} $$