$\sum \exp(-n^{ 1/3})$ series convergence

Question

$\sum_n \exp(-n^{ 1/3})$ Should be convergent by ratio test. However it gives me that $a_{n+1}/a_{n}=\exp(n ^{1/3}-(n+1)^{1/3}) \rightarrow 1,n \rightarrow +\infty $ so the limit ratio test fails. Thank you for suggestions.

Answer 1

HINT

Let apply limit comparison test with $\sum \frac 1{n^2}$.