Convergence, absolute convergence, divergence of series

Question

Let the series $\sum_{n=0}^\infty{\frac{a_n}{3^n}}$ be convergent, but the series $\sum_{n=0}^\infty{\frac{{(-1)^n}{a_n}}{3^n}} $ be divergent.
Show whether:

a)$\sum_{n=0}^\infty{\frac{a_n}{3^n}}$ is absolutely convergent or conditionally convergent

b)$\sum_{n=0}^\infty{\frac{a_n}{2^n}}$ is absolutely/conditionally convergent or divergent

c)$\sum_{n=0}^\infty{\frac{a_n}{4^n}}$ is absolutely/conditionally convergent or divergent

d)Find the convergence radius of $\sum_{n=0}^\infty{\frac{n+2}{n+3}a_nx^n}$

Okay, so far I have solved a), which I found quite easy to do, but I seem to get something wrong at b/c/d could someone please explain how they should be solved as it seems I have blocked completely...

Answer 1

HINT:

We have

$$\limsup_{n\to \infty}\sqrt[n]{\left|\frac{a_n}{3^n}\right|}=1$$

Hence, we must have

$$\limsup_{n\to \infty}\sqrt[n]{\left|a_n\right|}=3$$

Answer 2

Hint: The series $\sum \frac {a_n}{3^n}$ must cause the ratio/root test to fail. Otherwise, we couldn't have the conditional convergence as in a).