Power Series -- Convergence, Divergence, and Absolute Convergence

Question

Suppose that the power series $$\sum a_nx^n$$ is convergent at $x=-3$ and divergent at $x=5$. What can be said about the following:

1. convergence at $x=-2$ ?
2. absolute convergence at $x=2$ ?
3. convergence at $x=-6$ ?
4. convergence at $x=3$ ?
5. divergence at $x=-5$ ?

How can I go about getting the answers to each of these questions?

Answer 1

Denote $R$ the radius of convergence then we know that the given power series is

• convergent (absolutely) on the interval $]-R,R[$
• divergent for all $|x|>R$
• we can't conclude for $x=\pm R$ so the answer is $$T-T-F-?-?$$

Answer 2

Hint: The radius of convergence is some $R$ for $3 < R < 5$. This should settle questions 1,2,3 for you. For questions 4 and 5, consider $a_n = 1/(n3^n)$ vs. $a_n = 1/(n 4^n)$ and $a_n = 1/(n4^n)$ vs. $a_n = 1/(n5^n)$.