Is the series $\sum \frac{n+7^n}{n+5^5}$ converges or diverges I am asked to show whether the series


  
    $$\sum_{i=1}^{\infty}\frac{n+7^n}{n+5^n}$$
  


is convergent or divergent. I tried using Comparison Test but I was unsure of what to compare it to since there is addition in both the numerator and denominator. I also tried factoring out n from both parts of the fraction but this was not helpful either. I could use Ratio Test but we did not learn it yet. At this point I am stuck and have no idea what other methods to try.
Since$$\lim_{n\to\infty}\frac{n+7^n}{n+5^n}=\lim_{n\to\infty}\left(\frac75\right)^n\frac{n7^{-n}+1}{n5^{-n}+1}=\infty,$$your series diverges.
Apply limit comparison  with $$\sum _1^\infty (7/5)^n$$ which is a geometric series with  $r=7/5$.

Note that $$\lim _{n\to \infty} \frac {(\frac {n+7^n}{n+5^n})}{ (\frac {7}{5})^n} =1$$ 

Thus the series $$\sum_{i=1}^{\infty}\frac{n+7^n}{n+5^n}$$ diverges as well.