Answer
The radius of convergence for the power series is $R=0 $ and the interval of convergence is $10$.
Work Step by Step
Ratio Test; Let us consider an infinite series $\Sigma a_k$ with positive terms and suppose that $l=\lim\limits_{k \to \infty}|\dfrac{a_{k+1}}{a_k}|$
1) When $l \lt 1$. then series $\Sigma a_k$ converges absolutely.
2) When $l \gt 1$. then series $\Sigma a_k$ diverges.
3) When $l = 1$. then series $\Sigma a_k$ may converge or diverge, that is, the test is inconclusive.
Here, we have $a_k=k !(x-10)^k$
Now, $l=\lim\limits_{k \to \infty}|\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty}|\dfrac{(k+1) !(x-10)^{k+1}}{k !(x-10)^k}|=|x-10| \lim\limits_{k \to \infty} |k+1|=\infty$
So, the radius of convergence for the power series is $R=\dfrac{1}{l}=\dfrac{1}{\infty}=0$. This implies that the interval of convergence is $10$.
