Determining whether a monotonic sequence is increasing or decreasing

Question

From my notes from lecture:

${a_n}$ is increasing if $a_{n+1}-a_n>0$ and decreasing if $a_{n+1}-a_n<0$

When my professor worked through a problem, he explicitly left $a_{n+1}$ and $a_n$ on the same side of the inequality (as seen above).

So, for example, to show that ${a_n}={}\frac{n+1}{n}$ is decreasing….

$a_{n+1}-a_n=\frac{n^2+2n-n^2-2n-1}{n(n+1)}=\frac{-1}{n(n+1)}<0$

1) How can I know that $\frac{-1}{n(n+1)}<0$?

2) Is there are reason for leaving both a_n and a_{n+1} on the same side? Does it not make more sense to use $a_{n+1}>a_n$ and $a_{n+1}<a_n$?

EDIT: As a follow up question to my answer below, what about when $a_n$ includes $(-1)^n$?

Answer 1

Oh my...I just realized that $a_n$ is only for $n>1$. Therefore, $\frac{-1}{n(n+1)}$ has to be less than one.

Answer 2

You should not get hung up about what side the $a_n$'s are on. It matters what the sign is saying. That is for the case of monotone decreasing, $a_{n+1} < a_n$ is just the same as $a_n > a_{n+1}$.

Answer 3

1)You have a negative (and non-zero) numerator and an always positive denominator $(n\in\mathbb{N})$ so $-\frac{1}{n(n+1)}$ is always less than $0$

2)It's useful to use the definition

$a_{n+1}<a_n$

Depending on which is more convenient you get

$\frac{a_{n+1}}{a_n}<1$

and

$a_{n+1}-a_n<0$

Similarly for increasing sequences

3) As for your follow up question, think about how an alternating sequence can only converge if the non-alternating part converges to $0$