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?