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?

  • 2
    $\begingroup$ Number 2 corresponds to $x = 6$. $\endgroup$ – Daniel Fischer Apr 11 '18 at 21:11
  • 1
    $\begingroup$ What is x when (x-2)=4? $\endgroup$ – JB071098 Apr 11 '18 at 21:12

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