We are reading now about Cesàro summation. And there is a remark that:
We want $(a_n)$ and $(b_n)$ Cesàro-summable, then the Cauchy product $$(c_n)=\sum_{k=0}^na_kb_{n-k}$$ is Cesàro-summable. And that is not true, for example that does not hold for Grandi's series.
We are wondering that for elementary arithmetic operations ($+ - × ÷ $), did people ask the same question? We think if we have $(a_n)$ and $(b_n)$ Cesàro-summable then the sum $$(c_n)= (a_n + b_n)$$ should be Cesàro-summable, but we are not totally sure.
Could you please let us know if there is any book which wrote about that?
Thank you!
The book you are now reading should have already covered that Cesaro summation is linear:
Given Cesaro-summable sequences $(a_n), (b_n)$ and any constants $A, B$, the sequence $(Aa_n + Bb_n)$ is also Cesaro-summable, and further, $$\sum^C (Aa_n + Bb_n) = A\,\sum^C a_n + B\,\sum^C b_n$$ where I am using $\overset C\sum$ to denote Cesaro-summation.
Note that subtraction is covered in that formula by taking $A = 1, B = -1$.
In general it is not true that $(a_nb_n)$ or $\left(\frac {a_n}{b_n}\right)$ are Cesaro-summable just because $(a_n)$ and $(b_n)$ are. And even when they are Cesaro-summable, there is no reason to expect them to converge to the product or ratio of $\overset C\sum a_n$ and $\overset C\sum b_n$. This is only to be expected, as the same is true for ordinary series summation.