Final answer:
The question inquires about series expansions, particularly the remainder or error after n terms of the series. It's related to the calculation of the partial sum approximation of a series and how large a sample size might affect the accuracy of statistical measures like the mean.
Step-by-step explanation:
The question deals with the concept of series expansions and the approximation errors associated with using a partial sum to estimate the total sum of a series.
When we say that the error involved in using the partial sum sn as an approximation to the total sum s is represented by Rn = s - sn, and the size of the error is given by bn+1, we're discussing a scenario where we're truncating a series after n terms to approximate its sum.
For example, in the case of a binomial theorem expansion, (a + b)n can be expanded into several terms, and if we only take up to the nth term for calculation, there is an error involved in that partial sum which is related to the subsequent terms.
In practice, if the sample size n is large enough, certain statistical properties like the sample mean will approximate the population mean, and the distribution of sample means will approach a normal distribution through the Central Limit Theorem.
Therefore, calculating the sample size n accurately is crucial in statistics to ensure a specified margin of error or to predict the error in a series approximation.