Does $\sum\limits_{n = 1}^{\infty} \frac{3^n + 4^n}{2^n + 5^n}$ converge? My Attempt
First, check the limit
$$\lim_{n \to \infty} \frac{3^n + 4^n}{2^n + 5^n} = 
\lim_{n \to \infty} \frac{\left(\frac{3}{5}\right)^n + \left(\frac{4}{5}\right)^n}{\left(\frac{2}{5}\right)^n + 1} = 0.$$
So, we cannot conclude anything.
I used Comparison Test and Ratio Test.
Consider that
$$\frac{3^n + 4^n}{2^n + 5^n} < 2\frac{5^n}{2^n + 5^n}.$$
If I can proof the convergence of the series on the right-side, then it's done by comparison test.
I used ratio test, in order to proof the convergence of
$$\sum_{n= 1}^{\infty} \frac{5^n}{2^n + 5^n}.$$
$$\lim_{n \to \infty} \frac{55^n}{2(2^n) + 5(5^n)} \frac{2^n + 5^n}{5^n} = 5 \lim_{n \to \infty} \frac{2^n + 5^n}{2(2^n) + 5(5^n)} = 1.$$
I didn't find the way to proof.
Any suggestion? Thanks in advanced.
Solution (@abiessu & @Thomas Andrew)
Consider that
$$\frac{3^n + 4^n}{2^n + 5^n} < 2\frac{4^n}{2^n + 5^n}.$$
Proof this series converge.
$$\sum_{n= 1}^{\infty} \frac{4^n}{2^n + 5^n}.$$
Proof (Ratio Test)
$$\lim_{n \to \infty} \frac{44^n}{2(2^n) + 5(5^n)} \frac{2^n + 5^n}{4^n} = 4 \lim_{n \to \infty} \frac{2^n + 5^n}{2(2^n) + 5(5^n)} = \frac{4}{5}.$$
The series converge.
Hence
$$\sum_{n = 1}^{\infty} \frac{3^n + 4^n}{2^n + 5^n}$$
converge.

My preferred method would be to note that
$$
\frac{3^n+4^n}{2^n+5^n}
\leq \frac{4^n+4^n}{0+5^n}
= 2 \cdot \left(\frac{4}{5}\right)^n.
$$
So, the series converges by the comparison test and the geometric series criterion.
