You can post as many questions as you like, prefferably in different threads.
To check if a sequence is monotonic, look at . What can you say about the sequence if that expression is smaller than zero for all n, or greater than zero for all n?
How do you check if a sequence is monotonic? And then if it is increasing or decreasing? As example Un = n^3 − 2n^2
I really appreciate your help.
PS: Is there a problem if I post questions as soon as they come up, on a new thread? (I can't really afford to pay 100 euros for each book on calculus :\)
Well, it's hard for me to specifically tell if I don't know what you have learned and what not.
You can write (only if the sequence is differentiable) and find the derivative with respect to x. I can't recall anything else at the moment.
In your case, you can easily see that , so it is monotonic increasing.
The derivative of has 2 zeroes -- at 0 and 4\3. This means that it may have critical points in x-coordinates 0 and 4\3. It also means that for any , the function is monotonic. To check if it is increasing or decreasing, see what f'(2), for example, is. In this case, it is positive and so the sequence is monotonic increasing.
By the sequence being differentiable, I meant that the function , and in your case is differentiable -- otherwise you can't differentiate it and use the derivative.
Also, for cases like sequences with in them, the function won't even be well-defined (what is for a non-integer x?), and that can be troublesome!
If you haven't gone over differentiation yet, I suggest you wait for that and things will become clearer.