Calu said:Homework Statement
I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.
I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.
I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.Ray Vickson said:Well, PF rules require you to make a start on your own. What do you know about the magnitudes of ##\sqrt{n}## and ##\sqrt{n+1}##?
Calu said:I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.
A monotonically decreasing sequence is a sequence of numbers where each term is smaller than the previous one. In other words, the terms in the sequence are always getting smaller or staying the same.
To show that a sequence is monotonically decreasing, you must prove that each term in the sequence is smaller than the previous one. This can be done by using mathematical induction or by finding the difference between consecutive terms and showing that it is always negative.
No, a sequence cannot be both monotonically increasing and decreasing at the same time. If a sequence is monotonically increasing, it means that each term is larger than the previous one, while a monotonically decreasing sequence has each term smaller than the previous one. These two conditions cannot be satisfied at the same time.
A monotonically decreasing sequence can have consecutive terms that are equal, while a strictly decreasing sequence cannot. In other words, a strictly decreasing sequence is a monotonically decreasing sequence with no repeated terms.
No, not all monotonically decreasing sequences are convergent. A convergent sequence is one that has a limit as the number of terms approaches infinity. A monotonically decreasing sequence can converge or diverge, depending on the values of its terms.