if I have a series from n=3 to n=infinity, and the content is n*sin(1/n), what test can I do to see if it is absolutely convergent, conditionally convergent, or divergent.

A choice of test,
Divergence test
geometric series
p-series test
comparison test
limit comparison test
alternating series test
ratio test
root test
integral test

Just try the limit:

\lim_{n\to \infty}n\cdot sin(\frac{1}{n})=1\neq 0

So, the series diverges.

We can intuitively see this one diverges because of the product of sin with n.