Question

Infinite series e^(-n)sin(n) ? what test should I use?

Answer 1

Ok. Lets do this. e^(-n) can be rewritten as 1/e^n. So when we combine this with our sin(n) we get \[\sin (n)/e^n\]

Answer 2

I think you will need to use the Ratio Test.

Answer 3

Looks like a transient term, the series will converge.
e^-n will be a very small number

Integral test will conclude that it is a converging series.