Is there a general way to determine whether the nth partial sum of a series has an explicit, closed-form representation?

The $n$th term $a_n$ of any series has a closed form representation if the nth partial sum $S_n$ has a explicit, closed-form representation; specifically, $a_n = S_{n+1}-S_n$

I’m interested in the converse situation. If the $n$th term of a series has an explicit, closed-form representation, is the $n$th partial sum guaranteed to have an explicit, closed-form representation? If so, is there a general way to determine this?

When I say explicit, closed-form representation, I am referring to a non-recursive, analytic expression whose number of terms is finite and does not depend on $n$.