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: 
**Hint**. One may just observe that
$$0\leq \sum\_{n=2}^\infty \frac{n+2}{n^2+4n-7}\left(\frac{4}{7}\right)^n\leq \sum\_{n=2}^\infty \left(\frac{4}{7}\right)^n<\infty.
$$
**hint**: $\dfrac{n+2}{n^2+4n-7} < 1 (n \geq 3) $ and $\left|\dfrac{4}{7}\right| < 1$