I summed up $4$ kinds of series that I am having trouble solving. It seems like for the first one, the limit in infinity is - infinity which means it diverges.
The limit when $x \to \infty$ in 2 is $0$, so I think it converges.
Same goes for 3.
And I actually have no idea what I should do in 4.
So basiclly i'm stuck.
For the first two, as $n \to \infty, \frac 1n \to 0$, so $\sin \frac in \approx \frac 1n \text {and }e^{\frac 1n} \approx (1+\frac 1n)$ For the third you should know that exponentials dominate over polynomials. For the last, you are expected to plug in Stirling's formula for the factorials and see what comes out.