The discussion revolves around demonstrating that the sequence defined by \( a_n = \frac{1}{\sqrt{n}} \) is monotonically decreasing. Participants are exploring the necessary conditions to establish that \( a_n \geq a_{n+1} \).
Some participants have begun to articulate their reasoning regarding the inequality involving square roots, while others are seeking clarity on how to express their thoughts more coherently. There is an active exploration of the implications of the inequalities presented.
Participants are reminded of forum rules that encourage making an initial attempt before seeking help, which influences the nature of the discussion.
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.