# Power Series Convergence/Divergence

Suppose you have the power series $$\sum_{n=0}^{\infty}C_n(x-2)^n$$and it converges when $x=-1$ and diverges when $x=5$. Which of the following is true?

1. The radius of convergence is $R=3$

2. $\sum_{n=0}^{\infty}C_n(4)^n$ converges

3. $\sum_{n=0}^{\infty}C_n(2)^n$ converges

The correct answer is $1$ and $3$. I understand why $1$ is true, since if you have $|x-2| < 3$, solving it would give an interval of convergence $[-1,5)$. Why is $2$ false and $3$ true? Using the interval $[-1,5)$, wouldn't all be true?

• Number 2 corresponds to $x = 6$. – Daniel Fischer Apr 11 '18 at 21:11
• What is x when (x-2)=4? – JB071098 Apr 11 '18 at 21:12