Finding Upper and Lower Limits of Sn

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In summary, to compute the Upper and Lower limit of {Sn}, which is defined as S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + Sm, directly from its expression, you can look at each subsequence separately and observe the pattern in the formulas for the odd and even-subscript terms. This method does not require deduction of the terms.
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Ka Yan
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How can I compute the Upper and Lower limit of {Sn}, which defineded as: S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + Sm , directly from its expression, rather than by deduction of the terms?

(i.e., from the definition of Sn, instead of from 0, 0, 1/2, 1/4, 3/4, ...)

thks!

(I'm sorry, erm, I post this question here. I had moved it into "Precalculus Mathematics" of " Homework & Coursework Questions" .

Sorry, manager.)
 
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Ka Yan said:
How can I compute the Upper and Lower limit of {Sn}, which defineded as: S1 = 0, S2m = S2m-1 /2, S2m+1 = 1/2 + Sm , directly from its expression, rather than by deduction of the terms?

(i.e., from the definition of Sn, instead of from 0, 0, 1/2, 1/4, 3/4, ...)

thks!
Better late than never ...
Look at each subsequence separately, since there are different formulas for the odd-subscript terms and the even-subscript terms.
For example, for the odd terms,
##s_1 = 0##
##s_3 = \frac 1 2 + s_1##
##s_5 = \frac 1 2 + s_2 = \frac 1 2 + \frac 1 2 + s_0##
Continue the process until you see a pattern. Try a similar technique for the even-subscript terms.
 

What is the purpose of finding upper and lower limits of Sn?

The upper and lower limits of Sn provide important information about the variability and spread of a dataset. They can help determine the range of values within which most of the data falls, and identify any outliers or extreme values.

How do you calculate the upper and lower limits of Sn?

The upper limit of Sn is calculated by adding 1.5 times the interquartile range (IQR) to the third quartile (Q3) of the dataset. The lower limit is calculated by subtracting 1.5 times the IQR from the first quartile (Q1). This method is known as the Tukey's method or the 1.5*IQR rule.

What does the upper and lower limit of Sn tell us about the dataset?

The upper and lower limit of Sn provide information about the spread of the dataset. If most of the data falls within these limits, it indicates that the data is normally distributed and there are no significant outliers. If there are data points beyond these limits, it suggests that there may be extreme values or outliers in the dataset.

What is the significance of outliers in a dataset?

Outliers are data points that are significantly different from the rest of the dataset. They can occur due to measurement errors, natural variation, or other factors. Outliers can have a major impact on the results of statistical analysis, which is why it is important to identify and handle them properly.

Are there any limitations to using the 1.5*IQR rule for finding upper and lower limits of Sn?

Yes, there are some limitations to using the 1.5*IQR rule. It may not be suitable for datasets with extreme skewness or heavy-tailed distributions. In such cases, alternative methods such as the 3*IQR rule or the z-score method may be more appropriate. It is also important to consider the context and purpose of the analysis before determining the limits.

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