<s>[INST] Comparison test proof of convergence I want to prove that $\sum_{n=1}^\infty\frac{3^n+4^n}{3^n+5^n}$ converges. To do this, I said that $\sum_{n=1}^\infty\frac{3^n+4^n}{3^n+5^n}<3\sum_{n=1}^\infty (\frac{4}{5})^n$, which indeed converges. Hence by the comparison test, the series we are given converges. Is this valid?
 [/INST] Hint:
$$\frac{3^n+4^n}{3^n+5^n} \le \frac{3^n+4^n}{5^n} = \left(\frac 35 \right)^n + \left(\frac 45 \right)^n \le 2\left(\frac 45 \right)^n $$
</s>