I'm working on the following problem:
Find $\lim \sup_n a_n$ and $\lim\inf_n a_n$ of the sequence given by $a_n = 1 + (-1)^n\frac{2n+3}n.$
I've done the following work so far:
Let $\{i_n\}_n$ be the sequence given by $i_n = \inf\{a_k:k\ge n\}.$ Then, $i_n = \{-4, -2, \frac {-8}5, \frac{-10}7,\cdots\}.$
I know that the $\lim \inf_na_n = \lim_n i_n$, but I can't figure out how to take the limit of $i_n$.
Best Answer
Let's first notice that $\frac{2n+3}{n}$ converges to $2$. Therefore, the sequence $$(-1)^n\frac{2n+3}{n}$$
has only two limit points, which are $2$ and $-2$. This implies that $(a_n)$ has only two limit points, which are $3$ and $-1$. Therefore, you get $$\liminf a_n = -1 \quad \text{and} \quad \limsup a_n = 3$$