what are some of the methods to do divergent/convergent? I have no problem with integration finding volumes, surface area and such. But the whole series / sequence chapters completely throw me off, I have no idea where to even begin with problems given. What is the strategy for dealing with series/sequences? 

for example, I don't even know where to begin with this one:

\begin{equation} \sum_{n=1}^\infty \frac{1}{n^{\ln{n}}} \end{equation}
Sorry if this is discouraging, but basically the strategy is: trial and error.  You need to know things like the comparison test, ratio test, root test, integral test, alternating series test and any others you come across, but there is no definitive answer for which one to use in any particular case.  A couple of suggestions:


if the term $a_n$ includes factorials, the ratio test may be worth a try;
for rational functions the comparison test (perhaps in the limit form) might be good.


You also need to know the comparative rates of growth of various functions, starting with things like
$$\hbox{constant}&lt;\ln n&lt;n^\alpha&lt;a^n&lt;n!$$
for large $n$.

For the specific example you have asked, just note that $\ln n&gt;2$ for $n\ge8$ and then use a comparison test.
Comparison test can be used. Since $\ln n &gt; 2$ for $n\ge 9$ it follows $\displaystyle{\sum_{n=9}^{\infty}\frac{1}{n^{\ln n}}&lt;\sum_{n=9}^{\infty}\frac{1}{n^2}}$, which is a convergente series.