# 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.

• Hint: for convergent series, the terms must go to $0$. – lulu Nov 5 '19 at 21:19

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.

Note that

$$\frac{n+7^n}{n+5^n}\ge \frac{7^n}{2\cdot5^n}=\frac12\cdot \left(\frac 7 5\right)^n \to \infty$$

therefore the series can't converge indeed for any $$\sum_1^\infty a_n =L$$ we have necessarly that

$$S_{N+1}-S_N=\sum_1^{N+1} a_n-\sum_1^N a_n=a_{N+1} \to L-L=0$$

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.