When is $\limsup_{n \to \infty} (a_n+b_n) = \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$?

Question

I know that $$\limsup\limits_{n \to \infty} (a_n+b_n) \le \limsup\limits_{n \to \infty} a_n +\limsup\limits_{n \to \infty} b_n.$$ But what should apply to A and B if we should have "=" ?

I can't find anything about it in my notes.

Answer 1

As a previous commenter stated, it is sufficient that if either $\lim a_n$ or $\lim b_n$ exists, then equality holds. However, note that if $a_n=\sin(\tan(n))$ and $b_n=\tan(\sin(n))$, then $$\limsup(a_n+b_n)=\limsup(a_n)+\limsup(b_n)=1+\tan(1)$$

And neither limit as $n\to\infty$ exists. So the convergence of one of the sequences isn't a necessary condition to characterize this equality.