Five friends are leaving a bar for home. Aside from the owner of the car, one other person is deemed able to operate the vehicle
1 answer:
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Part 1: You can simplify to
Presumably, the sequence starts at <em>n</em> = 1. It's easy to see that the sequence is strictly decreasing, since larger values of <em>n</em> make either fraction smaller.
(a) So, the sequence is bounded above by its first value,
(b) And because both fractions in converge to 0, while remaining positive for any natural number <em>n</em>, the sequence is bounded below by 0,
(c) Finally, is bounded above and below, so it is a bounded sequence.
Part 2: Yes, is monotonic and strictly decreasing.
Part 3:
(a) I assume the choices are between convergent and divergent. Any monotonic and bounded sequence is convergent.
(b) Since is decreasing and bounded below by 0, its limit as <em>n</em> goes to infinity is 0.
Part 4:
(a) We have
and the (-1)ⁿ makes this limit alternate between -10 and 10. So the sequence is bounded but clearly not monotonic, and hence divergent.
(b) Taking the limit gives
so the sequence is unbounded and divergent. It should also be easy to see or establish that the sequence is strictly increasing and thus monotonic.
For the next three, I'm guessing the options here are something to the effect of "does", "may", or "does not".
(c) may : the sequence in (a) demonstrates that a bounded sequence need not converge
(d) does not : a monotonic sequence has to be bounded in order to converge, otherwise it grows to ± infinity.
(e) does : this is true and is known as the monotone convergence theorem.
The sample standard deviation is used to calculate the determine the spread of estimates for a set of observations (i.e., a data set) from the mean (average or expected value).
<h3>What is sample standard deviation?</h3>The spread of a data distribution is measured by standard deviation. The average distance between each data point and the mean is measured.
The sample standard deviation (s) is a measurement of the variation from the expected values and is equal to the sample variance's square root.
where
s = sample standard deviation
N = the number of observations
= the observed values of a sample item
= the mean value of the observations
Learn more about simple standard deviation, refer:
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