The following series with a polynomial as numerator and $e^n$ as denominator converge? The series is the following:
$$\sum_{n=1}^\infty \frac {8n^2-7}{e^n(n+1)^2} $$

You can also use the ratio test $$u_n=\frac {8n^2-7}{e^n(n+1)^2}$$ $$\frac{u_{n+1}}{u_n}=\frac{(n+1)^2 \left(8 (n+1)^2-7\right)}{e (n+2)^2 \left(8 n^2-7\right)}=\frac 1e \, \frac{8 n^4+32 n^3+41 n^2+18 n+1}{8 n^4+32 n^3+25 n^2-28 n-28}$$ the limit of which being $\frac 1e$.
