I'm trying to correct an old quiz. I want to see if I have sufficiently corrected the second problem. The first I am still a bit unsure what to do.
(1) Show that the following series is divergent if $\alpha \in \mathbb{R}$ such that $|\alpha|<1$.
$\sum_{k=1}^{\infty} \frac{1}{1+\alpha^k}$
The above I wasn't sure what to do so I left it blank.
(2) Use the root test to decide whether or not the following series converges: $\frac{1}{2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^4}+\frac{1}{2^5}+\frac{1}{3^6}+...$
First note: $a_n =\begin{cases} \frac{1}{2^n} \text{ if n is odd}\\ \frac{1}{3^n} \text{ if n is even } \end{cases}$.
So $\liminf a_n = \sqrt[2n]{\frac{1}{3^n}}=\frac{1}{3}$ and $\limsup a_n = \sqrt[2n-1]{\frac{1}{2^n}}=1$ So the sequence diverges. I just want to make sure I got the values correct on this one.