Question

How do I determine whether a sequence, given the nth term, diverges or converges?
Ex:
\[a_n =(-1)^n\frac{ n+1 }{ n+2 }\]

Answer 1

wiki has a list of tests
https://en.wikipedia.org/wiki/Convergence_tests
but it looks like your sequence approaches a sequence of +1 -1 +1 -1 ...
and it neither converges nor diverges

Answer 2

You can also use the limit test: If \(\lim\limits_{n\to\infty}a_n\not=0\) for any sequence \(a_n\), then the sequence diverges.