Find upper and lower bounds for n/m

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In summary, upper and lower bounds are numerical values used to estimate the true value of a given quantity. They represent the highest and lowest possible values, respectively. To find them for a given ratio, the range of values for the numerator and denominator must be determined and divided accordingly. These bounds are important in various scientific and mathematical fields as they provide educated guesses and help determine the margin of error. It is possible for the upper and lower bounds to be the same value, indicating an exact estimate. Factors such as measurement precision, assumptions, and data variability can affect the accuracy of these bounds.
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If n>m>1, find upper and lower bounds for n!/m!

Answer: upper bound = {[(n+1)^(n+1)]/[(m+1)^(m+1)]}*e^(-(n-m))
lower bound = [(n^n)/(m^m)]*e^(-(n-m))

This is from a chapter on finding bounds for sum of series. Can someone please explain how to arrive at the answer? Thanks!
 
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1. What is the definition of upper and lower bounds?

Upper and lower bounds are numerical values that serve as an estimate or approximation of the true value of a given quantity. The upper bound is the highest possible value, while the lower bound is the lowest possible value.

2. How do you find upper and lower bounds for a given ratio?

To find the upper and lower bounds for a given ratio, you need to first determine the range of values that the numerator and denominator can take. Then, divide the highest possible value of the numerator by the lowest possible value of the denominator to get the upper bound, and vice versa for the lower bound.

3. What is the importance of finding upper and lower bounds?

Finding upper and lower bounds is important in many scientific and mathematical fields as it allows us to make educated guesses about the true value of a quantity. It also helps in determining the margin of error in measurements and calculations.

4. Can upper and lower bounds be the same value?

Yes, it is possible for the upper and lower bounds to be the same value. This means that the estimated value is an exact value and there is no range of values in between for the given ratio.

5. What factors can affect the accuracy of upper and lower bounds?

The accuracy of upper and lower bounds can be affected by factors such as the precision of the measurements used, the assumptions made in the calculations, and the variability in the data. It is important to take these factors into consideration when determining the upper and lower bounds for a given ratio.

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