Answer
$\Sigma a_{n}$ is convergent.
Work Step by Step
Suppose that a series $\Sigma a_{n}$ has positive terms.
So $s_{n} - s_{n-1} = a_{n} \gt 0$ for all $n$
$\to$ $s_{n} \gt s_{n-1}$ for all $n$
Thus {$s_{n}$} is an increasing sequence.
Since $s_{n} \leq 1000$ for all of $n$
So {$s_{n}$} is bounded sequence. We know that every monotonic and bounded sequence is convergent.
So {$s_{n}$} converges, and the sequence of partial sums is convergent.
And then the series $\Sigma a_{n}$ is convergent.
