The definition of a function series

Question

I have the function series $f_n(x)$ defined as: $$ f_n(x) = \frac{xe^{-nx}}{\ln(n)} $$

where $ x\in \mathbb{R}~~ \& ~~ \forall n \geq 2 $.

The question askes for two things:

$1.$ to specify the domain where the series $ \sum_{n \geq 2} f_n(x)$ is defined (domain of definition).

$2.$ to study if the function is continues over the interval $]0, \infty[$.

I already solved the second question with no problem, but I don't know how to solve the first one! I would appreciate your help a lot.

Answer 1

If $x=0$, then you always have $f_n(x)=0$, and therefore the sequence converges.

If $x>0$, then $\lim_{n\to\infty}e^{-nx}=0$, and therefore $\lim_{n\to\infty}\frac{xe^{-nx}}{\log n}=0$ too.

Finally, if $x<0$, then $\lim_{n\to\infty}\frac{e^{-nx}}{\log n}=\infty$, and therefore $\lim_{n\to\infty}\frac{xe^{-nx}}{\log n}=-\infty$; in particular, your sequence diverges.