Answer
Sequence converges to $1$.
Work Step by Step
As we know that a sequence converges when $\lim\limits_{n \to \infty}a_n$ exists.
Consider $a_n=1+\dfrac{(-1)^n}{n}$
Apply limits to both sides.
$\lim\limits_{n \to \infty}a_n=\lim\limits_{n \to \infty}[1+\dfrac{(-1)^n}{n}]$
$\lim\limits_{n \to \infty}a_n=[\lim\limits_{n \to \infty}1+\lim\limits_{n \to \infty}\dfrac{(-1)^n}{n}]$
$\lim\limits_{n \to \infty}a_n=1+\dfrac{1}{\infty}$
$\lim\limits_{n \to \infty}a_n=1+0$
$\lim\limits_{n \to \infty}a_n=1$
Hence, the sequence converges to $1$.
