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.
$\endgroup$ 12 Answers
$\begingroup$$\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.
$\endgroup$ $\begingroup$Since \begin{align} \lim_{n\to\infty} \frac{\frac{1+5^{n+1}}{1+6^{n+1}}}{\frac{1+5^n}{1+6^n}} &=\lim_{n\to\infty} \frac{(1+5^{n+1})(1+6^n)}{(1+5^n)(1+6^{n+1})}\\ &=\lim_{n\to\infty} \frac{(5^{-n}+5)(6^{-n}+1)}{(5^{-n}+1)(6^{-n}+6)}\\ &=\frac{(0+5)(0+1)}{(0+1)(0+6)}\\ &=\frac{5}{6}<1, \end{align} ratio test works and given series converges.
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