In what range is $\sum_{n=1}^\infty \Big( \frac{x-2}{6} \Big)^n$ uniformly convergent?

Question

I have this multiple choice question:

The sequence $$\sum_{n=1}^\infty \Big(\frac{x-2}{6} \Big)^n $$ is uniformly convergent on the interval:

1. $-6 \leq x \leq 6$

2. $-3.8 \leq x \leq 7.8$

3. $-8 < x < 4$

4. $-4 < x < 0$

5. $x \in \mathbb{R}$

6. $-7.8 \leq x \leq 3.8$

I have trouble solving this question, because I don't know how to determine the range of uniform convergence. I looked up Abel's uniform test for convergence, but this is "only" for determining if a sequence is uniformly convergent, not for determining the range. I hope someone can help me with this.

Answer 1

We need $|\dfrac{x-2}6|\lt1\implies |x-2|\lt6\implies -4\lt x\lt8$.

Thus the answer is $2)$, as we need to restrict ourselves to a compact subinterval.