# Thread: How to check if a sequence is monotonic?

1. ## How to check if a sequence is monotonic?

Hello everyone.

How do you check if a sequence is monotonic? And then if it is increasing or decreasing? As example Un = n^3 − 2n^2

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 :\)

2. You can post as many questions as you like, prefferably in different threads.

To check if a sequence is monotonic, look at $U_{n+1} - U_{n}$. What can you say about the sequence if that expression is smaller than zero for all n, or greater than zero for all n?

3. Well I assume if it's greater than 0, it's increasing, if smaller then it is decreasing?

EDIT: How do I know if it's monotonic at all?

4. Well, it's hard for me to specifically tell if I don't know what you have learned and what not.

You can write $f(x) = U_x$ (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 $U_{n+1} - U_n \geq 0 \ \forall n \geq 1$, so it is monotonic increasing.

5. Originally Posted by Defunkt
Well, it's hard for me to specifically tell if I don't know what you have learned and what not.

You can write $f(x) = U_x$ (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 $U_{n+1} - U_n \geq 0 \ \forall n \geq 1$, so it is monotonic increasing.
And what does the derivative result mean? For the $U_n = n^3 - 2n^2$
case you get $3n^2 - 4n$

Also can you teach me what/when is a sequence differentiable?

Thanks alot!

6. The derivative of $U_n$ 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 $x\geq \frac{4}{3}$, 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 $f(x) = U_x$, and in your case $f(x) = x^3 -2x^2$ is differentiable -- otherwise you can't differentiate it and use the derivative.
Also, for cases like sequences with $n!$ in them, the function won't even be well-defined (what is $x!$ 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.