Calculating $\limsup\limits_{n \to \infty}a_n$ and $\liminf\limits_{n \to \infty}a_n$ with $a_n=\left( \frac{n+(-1)^n}{n} \right)^n$ I need to calculate $\limsup\limits_{n \to \infty} a_n$ and $\liminf\limits_{n \to \infty}a_n$ where $a_{n} = \left( \dfrac{n+(-1)^n}{n}\right)^n, \: n \in \mathbb{N}$
My approach is to calculate the limits for $n$ being even or odd.


*

*$n \pmod 2 \equiv 1$


$$\quad \lim_{n \to \infty} \left(\dfrac{n-1}{n} \right)^n$$


*

*$n \pmod 2 \equiv 0$


$$\quad \lim_{n \to \infty} \left(\dfrac{n+1}{n} \right)^n$$
I don't know how I can solve the two formulas above. How should I continue?

Hint. One may recall that
$$
\lim_{n \to \infty} \left(1+\frac{x}n\right)^n=e^x, \qquad x \in \mathbb{R},
$$ and one may observe that
$$
\left(1-\frac1n\right)^n\le a_n \le \left(1+\frac1n\right)^n, \quad n=1,2,\cdots.
$$
