In general if $$S={\Sigma}_n^\infty a_n $$
converges, under what circumstnces can a convergence exist for partial/truncated series sum :
$$ Sn=\Sigma_1 ^n a_k, \quad k\lt n ?$$
Also can there be a closed form expression for this partial series?
I am not quite familiar in the area, a short guiding comment/answer would suffice. Thanks in advance.