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?