How to find the limit superior of $(-1)^n + \frac{1}{n}$?

Question

If I have a sequence defined as $a_n = (-1)^n + \frac{1}{n}$, how would I find the limit superior?

Answer 1

Odd terms are non-positive, while even terms are greater than $1$. Hence the limit superior is also the limit superior of the even terms – which converge to $1$. Thus $$\limsup_{n\to\infty} a_n=1.$$