Determine the convergence of the series $\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$ Does 
$$\sum_{n=1} ^{\infty} \frac{5^{n}-2^{n}}{7^{n}-6^{n}}$$
converge?
I tried the ratio test but I failed. 

Ratio test: 
$$
\frac{\displaystyle\frac{5^{n+1}-2^{n+1}}{7^{n+1}-6^{n+1}}}{\displaystyle\frac{5^{n}-2^{n}}{7^{n}-6^{n}}}
=\frac{7^{n}-6^{n}}{{7^{n+1}-6^{n+1}}}\,\frac{5^{n+1}-2^{n+1}}{5^{n}-2^{n}}
=\frac{1-(6/7)^n}{7-(6/n)^{n+1}}\,\frac{5-(2/5)^{n+1}}{1-(2/5)^n}\to\frac57<1,
$$
so the series converges. 
