[tex]\displaystyle\sum_{k\ge0}\frac k{3^k}(x-6)^k[/tex]
converges by the ratio test for
[tex]\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<1[/tex]
The limit is 1, so the series converges as long as
[tex]\dfrac{|x-6|}3<1\implies|x-6|<3[/tex]
which indicates a radius of convergence of 3.
