How can I determine if this series converges or diverges?

Question

My series is (the sum from n=1 to infinity) e^(-nx) I am try to determine if it diverges or converges when 0

Answer 1

modulo function gave the right answer, but his last sentence may be confusing, so I have tried to write it more clearly here:

So it will converge for e^-x < 1.
This will be true if: -x < 0.
Multiplying both sides of this inequality by -1 and changing the sense of the inequality, we have: x > 0.

So, the series will converge if x > 0.

Answer 2

Let r = e^(-x),
then your series is the geometric series: sum 1 to inf of r^n, right? This converges
for r < 1. So it will converge for e^-x < 1 for -x < 0 or x > 0.

Answer 3

Ok, I think the real trick here is to separate

e^-nx = (e^-x)^n. Now you can apply the other convergence tests like the ratio. Also does it look similar to the geometric series? Try proof/disprove by comparison.