Does $\sum_{n=1}^\infty \frac{n+2}{n^2+4n-7}\left(\frac{4}{7}\right)^n $ converge?

Question

Does the following series converge, does it converge absolutely? $$\sum_{n=1}^\infty \frac{n+2}{n^2+4n-7}\left(\frac{4}{7}\right)^n $$

Now I thought about using Dirichlets Test of Uniform Convergence.

I know that $|\sum_{n=1}^\infty (\frac{4}{7})^n|\leq M < \infty$.

I am left to show that if $\frac{n+2}{n^2+4n-7} \rightarrow 0$ uniformly then my whole series converge uniformly?

What does uniform convergence of series mean?

I know that it means partial summs converge uniformly, but what is it in lazy mans terms? Does it imply absolute convergence?

Answer 1

Hint. One may just observe that $$0\leq \sum_{n=2}^\infty \frac{n+2}{n^2+4n-7}\left(\frac{4}{7}\right)^n\leq \sum_{n=2}^\infty \left(\frac{4}{7}\right)^n<\infty. $$

Answer 2

hint: $\dfrac{n+2}{n^2+4n-7} < 1 (n \geq 3) $ and $\left|\dfrac{4}{7}\right| < 1$