Mathematics

Problem

Consider the series

\displaystyle\sum_{k=1}^{\infty}\frac{(x-1)^{2k}}{k\,4^{k}}

. Which of the following correctly states its radius of convergence and the behavior of the series at

x=3

? a) The radius of convergence is

1

, and the series diverges at

x=3

. b) The radius of convergence is

2

, and the series diverges at

x=3

. c) The radius of convergence is

2

, and the series converges conditionally at

x=3

. d) The radius of convergence is

4

, and the series converges absolutely at

x=3