Does the series $\sum_{n=1}^{\infty} \frac{19+(n+5)!}{(n+7)!}$ Converge or diverge.

Question

Does the series $\displaystyle\sum_{n=1}^{\infty} \dfrac{19+(n+5)!}{(n+7)!}$ Converge or diverge

I tried using the ratio test, but it gave me this which I don't see how you'd be able to solve. $$\lim_{n\to\infty} |\frac{a_{n+1}}{a_{n}}| = \lim_{n\to\infty}\frac{19+(n+6)!}{19+(n+5)!*(n+8)}$$

Wolfram alpha told me to use the comparison test, but I can't for the love of god see which series I'd compare it to.

Answer 1

$$\sum_{n=1}^{\infty} \frac{19+(n+5)!}{(n+7)!}=19\sum_{n=1}^{\infty} \frac{1}{(n+7)!}+\sum_{n=1}^{\infty} \frac{1}{(n+7)(n+6)}$$ and $$ \frac{1}{(n+7)!}<\dfrac{1}{n^2}~~~,~~~\frac{1}{(n+7)(n+6)}<\dfrac{1}{n^2}$$

Answer 2

$$\sum_{n=1}^\infty\frac{19+(n+5)!}{(n+7)!}=\sum_{n=1}^\infty\frac{1+\frac{19}{(n+5)!}}{(n+6)(n+7)}<\sum_{n=1}^\infty\frac2{n^2}$$