Q:

Where am I making my mistake? ( intervals of convergence)

The sum is:
\begin{align}
\sum_{n=1}^{\infty} \frac{(x-6)^n}{(-8)^nn}
\end{align}
I end up with $|(x - 6)/8| < 1$ and therefore, $-8 < 6 - x < 8$ so $14 > x > -2$, but that gives me the wrong answer.   
I've been up all night doing this homework so chances are I'm missing something obvious or committing a rookie mistake.  Thank you for your help.  

A:

You have derived $-8<6-x<8$. Subtracting $6$ from each term gives $-14<-x<2$. Now negating all the terms gives $-2<x<14$. 
So now we check if the sum converges at the boundary points of the interval.  When $x = 14$, the sum is $\sum \frac{(-1)^n}{n}$, which converges.  When $x = -2$, the sum is $\sum \frac{1}{n}$, which diverges.  The sum should then converge on $-2<x\leq 14$.

