Interval of convergence for a power series

[petrol.veem]

I'm trying to find out whether the "edges" of the following power series are within the interval of convergence. What I have so far...

Sigma (n=0,inf) [ (x-2)^n / 10^n ]

So using the root test, and after a bit of work, I get:

-8 < x < 12

Now I am a little bit confused about what happens at x=-8 and x=12

For x=-8, the series becomes Sigma(n=0,inf) [ (-1)^n ] which diverges.

And for x=12, the series becomes Sigma(n=0,inf) [ 1^n ] which converges.

However, the book also asks me to find the sum of the series as a function of x. Clearly its a geometric series, so its sum would be:

1 / ( 1 - (x-2)/10 ) but this is not defined at x=12

So now I am not sure if I should write the interval of convergence as (-8,12) or (-8,12]

 

[pka]

It is fairly clear that the series does not converge at either endpoint,
\(\displaystyle {\frac {(-8 -2)^n} {10^n} }= {(-1)}^n\)
\(\displaystyle {\frac {(12 -2)^n} {10^n}} = {(1)}^n\)