Methods to prove a series is decreasing? What are soome good methods to show a series is decreasing?
Usually I just compare the $a_n$ and $a_{n+1}$ values. For example $\frac 1 {k^2} >\frac 1 {k^2+1}$ so it shows the series is decreasing. I heard you can also do something like $\frac{a_{n+1}}{a_n}$ but that seems complicated.
What are some more efficient methods for like series that have $\frac{k+2} {k(k+3)}$?

If you are more comfortable with things like $\frac{1}{k^2} > \frac{1}{k^2 +1}$ then another approach for cases like $\frac{k+2}{k(k+3)}$ is to use partial fraction decomposition:
$$\frac{k+2}{k(k+3)} - \frac{k+3}{(k+1)(k + 4)} = \left(\frac{1/3}{k + 3} + \frac{2/3}{k}\right) - \left(\frac{1/3}{k + 4} + \frac{2/3}{k+1}\right) = \frac{1}{3}\left(\frac{1}{k+3} - \frac{1}{k+4}\right) + \frac{2}{3}\left(\frac{1}{k} - \frac{1}{k+1}\right)$$
