Decide whether the infinite series converges of diverges.
$$\sum\_{n=1}^\infty \frac{2^n+3^n}{3^n+4^n}$$

My thought process:   

 The nth term test doesn't seem viable after initial use.   

 The comparison function I derived for DCT/LCT is: $\frac{3^n}{4^n}$.
 I cannot create an appropriate inequality between the given function and my comparison function I found, so I tried LCT. I am aware that 
$\sum\_{i=1}^\infty \frac{3^n}{4^n}$ is geometric and because the common ratio (r) is $ \frac{3}{4}$, $-1<r<1$ the series converges, however I cannot compute the limit LCT requires.   

Application of the root test would seem unbeneficial because roots do not split over sums.   

As for the ratio test I was also unsuccessful in proceeding with the computation of the limit.   

The above are the only tests I have learned as of right now. My question is what test should I be looking to utilize to determine convergence or divergence?