All limit points of $\{(-1)^k+(-1/k)^k\}$

Question

consider the sequence $\{x_{k}=(-1)^k+(-1/k)^k: k+1, 2, ...\}$

I know that $ \liminf_k \ x_k =-1$ and $ \limsup_k \ x_k =1$.

My question: is there any other limit point for this sequence? why or why not?

Thanks in advance.

Answer 1

Hint: Let $c\neq \pm1$ be a third limit point. That would mean that $\forall \epsilon > 0: |x_i-c|\leq \epsilon$ infinitely often.

If $|c|<1$ is there an $\epsilon>0$ that $|x_i-c|$ will not drop below. If $|c|>1$, is there an $N: |x_i - c|<\frac12(|c|-1)\;\forall i>N$