How to prove Convergence of this Series

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1. Feb 28, 2016

Euler2718

1. The problem statement, all variables and given/known data

Use any appropriate test to determine the convergence or divergence of the following series:

$$\sum_{i=0}^{\infty} \frac{2^{i} + 3^{i}}{4^{i}+5^{i}}$$

2. Relevant equations

3. The attempt at a solution

I've run it through mathematica and it told me it's convergent. However, I can't seem to find the right test to use. Ratio test / Limit comparison doesn't seem to work as nothing cancels, I can't find a test series for direct comparison, and root test wouldn't work? Have I over looked anything?

2. Feb 28, 2016

Ssnow

You can start with an asymptotic approach $\sum_{i=0}^{\infty}\frac{2^{i}+3^{i}}{4^{i}+5^{i}}\sim \sum_{i=0}^{\infty}\frac{3^{i}}{5^{i}}$

3. Feb 28, 2016

Samy_A

Try comparison test.
Hint: if 0<a<b and 0<c, 0<d, then $\frac{a+b}{c+d}\leq \frac{2b}{d}$

4. Feb 28, 2016

WWGD

How about : $0<a<b<c<d$ then $\frac {a+b}{c+d}$?

5. Feb 28, 2016

Euler2718

Alright I think I got it now, thanks.