Arbogast's "No rigorous convergence methods"? (soft question)

Question

It notes in the late 1800s history of Louis François Antoine Arbogast, who is notable for Faa di Bruno's formula 55 years earlier, that he said there was no rigorous method to prove the convergence of series. It looks like there a lot of divergence methods and lots of ways to analyze convergence by the usual analysis of comparing terms with limits, which seems rigorous. Did he mean for cases like uncountable sets? Not sure if this was just a product of its time or does math today have no rigorous method to provide convergence? For example if it was possible to compound every possible divergence/convergence method available today would it be possible to prove that any series converged?