I need to show whether $\displaystyle\sum_{n=1}^{\infty}{\frac{2^n+3^n}{4^n-5^n}}$ converges or diverges using the ratio test.
So far I have $\dfrac{a_{n+1}}{a_n} = \dfrac{2^{n+1}+3^{n+1}}{4^{n+1}-5^{n+1}} . \dfrac{4^n-5^n}{2^n+3^n}$
I know I could maybe use division by $5^n$ at some point but I am not sure how I could simplify from here in order to obtain an expression that I could easily show has a limit $<1$ for convergence or $>1$ for divergence.
The only tests I can utilise at the moment are ratio and comparison, along with the use of geometric series. I would easily be able to solve this if not for the minus sign in the denominator by using comparison with geometric series for example.