convergence of series : rational with exponentials

Question

I have this infinite series below, not sure how to handle the alternating term in the bottom. I am guessing i could use triangle inequality on the bottom, then the absolute of the alternating term would become positive. Not sure if i can replace or remove this term. Need some help

$$ \sum \frac{7^n}{(-2)^n +5^n} $$

Answer 1

The simplest way would be to notice that $$ 5^n + (-2)^n = 5^n + (-1)^n\cdot 2^n \leq 5^n + 2^n \leq 5^n + 5^n = 2\cdot 5^n $$ for every $n\geq 1$, and therefore $$ \sum_{n=1}^N \frac{7^n}{5^n + (-2)^n} \geq \sum_{n=1}^N \frac{7^n}{2\cdot 5^n} = \frac{1}{2}\sum_{n=1}^N \left(\frac{7}{5}\right)^n\,.$$ Now, for $N\to\infty$, the RHS diverges to $\infty$, so...

Answer 2

It suffices to observe that

$$a_n=\frac{7^n}{(-2)^n +5^n}\ge \frac{7^n}{2^n +5^n}=\frac{(2+5)^n}{2^n +5^n}\ge1$$

and therefore $$\sum_{n=1}^N a_n\ge N \to \infty$$