Investigate for convergence or divergence

$\require{cancel}$Investigate for convergence or divergence:

$$\sum_{i=1}^\infty \frac{3^n+4^n}{4^n+5^n}$$

I'm allowed to use basic tests for convergence or divergence:

• P-series test
• Geometric series test
• n^th Term test
• Integral test
• Comparison test
• Limit Comparison test
• Ratio test
• Root test
• Alternating test

The bold items are the ones I think are more likely to be used in this problem as opposed to the others.

Progress

So I made some progress but I'm not sure how to proceed.

I assume,$$a_n=\frac{3^n+4^n}{4^n+5^n}$$and,$$b_n=\frac{\cancel{3^n}+4^n}{\cancel{4^n}+5^n}=\frac{4^n}{5^n}=\left(\frac{4}{5}\right)^n$$The reason I cancelled the $3^n$ and the $4^n$ is because they're insignificant when compared to $4^n$ and $5^n$, respectively.

At this point, I know that $\sum_{i=1}^\infty b_n$ converges, because it is a geometric series. But I'm not sure how to proceed from here.