# Remainder estimate for product series

• bruno67
In summary, we have two series A and B with estimates for the remainders R_A^N and R_B^N. We are interested in finding an estimate for the remainder R_C^N of the product series AB. It is possible to derive an estimate for R_C^N using the remainders for A and B, given by the formula R_C^N = A_{N-1} R_B^N + B_{N-1} R_A^N.
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 $B$?

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?

## What is a remainder estimate for product series?

A remainder estimate for product series is a mathematical technique used to approximate the value of a series, or sum, of numbers that are multiplied together. It is used to determine how close the approximation is to the actual value of the series.

## How is a remainder estimate for product series calculated?

A remainder estimate for product series is calculated by taking the difference between the value of the series and the value of the approximation. This difference is then compared to a predetermined value, known as the error bound, to determine the accuracy of the approximation.

## Why is a remainder estimate for product series important?

A remainder estimate for product series is important because it allows us to determine the accuracy of an approximation and to improve the approximation if necessary. It also helps us understand the behavior of the series and make predictions about its future values.

## In what situations would a remainder estimate for product series be used?

A remainder estimate for product series is commonly used in calculus and other areas of mathematics where series are encountered. It can also be applied to real-world problems, such as financial forecasting or population growth, where the series can be modeled using mathematical equations.

## Are there any limitations to using a remainder estimate for product series?

While a remainder estimate for product series can provide valuable information about the accuracy of an approximation, it is important to note that it is only an estimate and may not always be completely accurate. Additionally, it may be more difficult to calculate a remainder estimate for series that involve large numbers or complicated functions.

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