Answer
The sequence diverges
Work Step by Step
Our inference, based on the graph (see below)
is that the sequence diverges.
Analytically, we base our conclusion applying the definition:
"The limit of a sequence $\{a_{n}\}$ is $L$, written as $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$
if for each $\epsilon > 0$, there exists $M > 0$ such that $|a_{n}-L| < \epsilon$ whenever $n > M$."
Re-wording the definition:
If, for any small interval around L, $(L-\epsilon,L+\epsilon)$
we can find the M-th term
after which all terms belong to that interval,\
then L is the limit of $\{a_{n}\}$
Since $\{a_{n}\}=\{1,0, -1,0,1, \ldots\},$
we can find an $\epsilon$, for example $\epsilon=0.1$, for which
there does not exist an M-th term
after which all terms will be within $\pm 0.1$ of any chosen number L.
So the "for each $\epsilon > 0$, there exists $M > 0$ such that..." is not satisfied.
The sequence does not have a limit.
The sequence diverges
