Determine $a_n :=1- \frac{(-1)^n}{\sqrt{n}}, \lim \inf a_n,\lim \sup a_n, \lim a_n$

Question

I am to determine $a_n :=1- \frac{(-1)^n}{\sqrt{n}}, \lim \inf a_n,\lim \sup a_n, \lim a_n$

I have checked the very first elements:

$n = 1: 1- \frac{(-1)^1}{\sqrt{1}} =1--1 =2$

$n = 2: 1- \frac{(-1)^2}{\sqrt{1}} =1-\frac{1}{\sqrt{2}} \approx 0.707$

$n = 3: 1- \frac{(-1)^3}{\sqrt{3}} =1--\frac{3}{\sqrt{3}} \approx 1.577$

$n = 4: 1- \frac{(-1)^4}{\sqrt{4}} =1-\frac{1}{2} = 0.5$

My intuation says that I have to check two subsequentes separated, where $n=2$ or to the even case $n=2k+1$.

I guess that I should check out that $a_n$ is positive for all $n \in \mathbb{N}$ which could give us a bound but I'm still struggling at this point.

Answer 1

Since $\frac{(-1)^n}{\sqrt{n}}$ tends to zero as $n\to\infty$ (its absolute value is $\frac{1}{\sqrt{n}}$), the sequence $a_n$ converges to $1$ as $n\to\infty$ so its $\liminf,\limsup$ and limit are all equal to $1$.