Remainder estimate for product series

[bruno67]

Suppose I have two series

$A=\sum_{n=0}^\infty a_n$

$B=\sum_{n=0}^\infty b_n$

and I have estimates for the remainders of each one:

$\sum_{n=N}^\infty a_n \le R^N_A$

$\sum_{n=N}^\infty b_n \le R^N_B$

Consider the product series

$AB=\sum_{n=0}^\infty c_n$

where $c_n=\sum_{i=0}^n a_i b_{n-i}$. Is it possible to derive an estimate for the remainder of $C$ based on the ones for $A$ and $B$?

 

[I like Serena]

Hi bruno67!

Let's define $A_N = \sum\limits_{n=0}^N a_n$ and $B_N = \sum\limits_{n=0}^N b_n$.

After writing out your formulas, I found I can write your remainder for C as:
$$R_C^N = A_{N-1} R_B^N + B_{N-1} R_A^N$$

Is that what you're looking for?