Convergence of $\sum\limits_{m= 0}^{\infty} \frac{(-3)^m+2^m}{3^m+2^m}$

Question

I'm trying to determine whether the series $$\sum\limits_{m= 0}^{\infty} \frac{(-3)^m+2^m}{3^m+2^m}$$ is absolutely or conditionally convergent or divergent. I've tried applying the comparison and ratio test to prove that it's not absolutely convergent without success. I've also been unable to apply the Leibniz test to prove that the series itself if divergent. My intuition tells me the series is divergent but I'm not sure how to prove it.

Answer 1

Write the numerator as $(-3)^m + (-2)^m -(-2)^m +2^m =(-1)^m((3)^m + (2)^m) +2^m(1-(-1)^m) $.

The terms become $(-1)^m+$ a convergent sum, so the sum oscillates and is therefore divergent.