Remainder estimate for product series

  • Thread starter bruno67
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  • #1
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Suppose I have two series

[itex]A=\sum_{n=0}^\infty a_n[/itex]

[itex]B=\sum_{n=0}^\infty b_n[/itex]

and I have estimates for the remainders of each one:

[itex]\sum_{n=N}^\infty a_n \le R^N_A[/itex]

[itex]\sum_{n=N}^\infty b_n \le R^N_B[/itex]

Consider the product series

[itex]AB=\sum_{n=0}^\infty c_n[/itex]

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

Answers and Replies

  • #2
I like Serena
Homework Helper
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Hi bruno67! :smile:

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

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

Is that what you're looking for?
 

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