Question

(k/3^k)(x-6)^k

Answer 1

$\displaystyle\sum_{k\ge0}\frac k{3^k}(x-6)^k$

converges by the ratio test for

$\displaystyle\lim_{k\to\infty}\left|\frac{\frac{k+1}{3^{k+1}}(x-6)^{k+1}}{\frac k{3^k}(x-6)^k}\right|=\frac{|x-6|}3\lim_{k\to\infty}\frac{k+1}k$

The limit is 1, so the series converges as long as

$\dfrac{|x-6|}3$

which indicates a radius of convergence of 3.