<s>[INST] Does $\sum_{n=1}^\infty \frac{1+5^n}{1+6^n}$ converge? Question 
Use appropriate tests to decide whether the series converges or diverges
$$\sum_{n=1}^\infty \frac{1+5^n}{1+6^n}$$
I'm not sure how to complete this question. I have attempted to complete this with the ratio test, however this becomes very messy quickly.
If it is the right method, can someone show me how to do it, otherwise how would you figure this out?
$$\lim_{n\to\infty} \left(\frac{1 +5^{n+1}}{1+6^{n+1}} \frac{1+6^n}{1+5^n}\right)$$
 [/INST] It is not that messy...
$$\frac{a_{n+1}}{a_n} = \frac{1+5^{n+1}}{1+6^{n+1}}\frac{1+6^{n}}{1+5^{n}} = \frac{1+6^{n}}{1+6^{n+1}}\frac{1+5^{n+1}}{1+5^{n}} = \frac{6^n(6^{-n}+1)}{6^n(6^{-n}+6)}\frac{5^n(5^{-n}+5)}{5^n(5^{-n}+1)} = \frac{6^{-n}+1}{6^{-n}+6}\frac{5^{-n}+5}{5^{-n}+1}.$$
As $n\rightarrow\infty$, $6^{-n},5^{-n}\rightarrow 0$, and thus the limit of the ratio test is $\frac{5}{6}$. Conclude that ...
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