Series Convergence Help!

$\Sigma ln(\frac{n}{n+1}) -ln(\frac{n+2}{n+3})$


The sum is from 1 to infinity.

Usually I approach series problems by testing to see if it diverges first but after applying the L'H rule, the series does approach 0 as it goes to infinity.

Usually from here I try the comparison/ratio/root tests. I'm guessing that the comparison test can work here in some way although I'm not able to see which series to compare it to?

Any idea how to prove this series diverges?

I got that, instead I converted both though to a single log series and took the limit from there. My lecturer tells me there's a comparison from there which can prove it diverges. I'm wondering which one it is?

The reason why I wrote it like that is because it makes it a little easier to see what's going on:

$\displaystyle\sum_1^m \ln \left(\dfrac{n}{n+1}\right)-\ln \left(\dfrac{n+2}{n+3}\right) =-\ln 3-\ln (m+1)+\ln (m+3)$

Which obviously converges to $\ln \frac{1}{3}$

Thanks!.

One last series if you could please.

$\displaystyle\sum 4^n\left(2+\sin n - \frac{1}{2+\sin n}\right)$

Well, does the summand converge to 0?