Do all series have a closed form representation of their partial sum? If not, can we feasibly prove that this is not the case?

The question was motivated by the way in which we approach the convergence and divergence of some series.

During my undergraduate analysis course one of the only times in which the partial sum was used to prove the convergence of some series was in the case of the geometric series. There we took the limit of the partial sum to deduce the convergence of the geometric seriese for some values.

However after that one proof we proceeded to knockout the classic tests for convergence e.g majorant/minorant, root test, etc.

So the question is, for all series which we have had to use some test to deduce their convergence or divergence, do they have closed form partial sums? If there exists series which we can deduce their convergence or divergence via some test, but we do not know of any closed form OR there quite simply does not exist a closed form, could an example or proof of this be given?