# Decide whether the series ${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$ converges or diverges

Determine whether the series converges or diverges $${\sum_{n=1}^{\infty} \frac{1+5^n}{1+6^n}}$$

I was thinking I should use ratio test but I get an ugly sequence that I don't know how to evaluate. Also, the only tests we can use are comparison test, ratio test, alternating series test, divergence test and integral test and I can't seem to find one that works.

• A ratio test should handle this. $\qquad$ – Michael Hardy May 21 '16 at 3:24

$\dfrac{1+5^n}{1+6^n}\leq \dfrac{1+5^n}{6^n}\leq 2\left(\dfrac{5}{6}\right)^n$. Now use the comparison test.