Limit of a sum with bound depending on the limit

Question

I am getting confused in calculating:

$\limsup_{n \to \infty} \sum_{k=n}^{\infty} a_k $, with $ a_k \xrightarrow{k \to \infty} 0$ and $0 \leq a_k \leq 1$ for each $k$.

a) Just using the easy bound:

$\limsup_{n \to \infty} \sum_{k=n}^{\infty} a_k \leq \limsup_{n \to \infty} \sum_{k=n}^{\infty} 1 $

I am not sure which of the following then holds (I guess the first but I cannot understand why the second is wrong):

1. $\limsup_{n \to \infty} \sum_{k=n}^{\infty} 1 = \limsup_{n \to \infty} ( \infty - n) = \infty $
2. $\limsup_{n \to \infty} \sum_{k=n}^{\infty} 1 = \limsup_{n \to \infty} \sum_{k=n}^{n} 1 = 0 $

b) If, as I guess, 1) holds, and as the lower index of the sum is depending on n, I cannot exchange limit and sum using monotone or dominated convergence. Has anybody an idea on how to deal with that?

Any hints or ideas would be greatly appreciated.