Question

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

for this series determine whether it is convergent or not .If convergent find radius of convergence

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{(k+1)/(3^(k+1))(x-6)^(k+1)}{\frac k{3^k}(x-6)^k}\right|=\fracx-63\lim_(k\to\infty)\frac{k+1}k<1`

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

`\frac3<1\implies|x-6|<3`

which indicates a radius of convergence of 3.