One of the hypotheses of the Abel's test for product series convergence is stronger than the corresponding Dirichlet's test; that is, the former imposes the convergence of one of the series and the latter only requires the partial sum sequence of the series to be bounded; hence the Dirichlet's test is more general than the Abel's test. Then I wonder what is a possible reason for the Abel's test to stay in introductory textbooks.
One reason I can think of is that it is due to the name "Abel", like the fact that the name "Riemann" still lives in integration teaching in spite of that Riemann integration has become a very special case.
However, please take not this question wrong; I am just wondering if there is any deeper consideration (maybe a pedagogical one) I overlooked.
Although saying this may be redundant, but: I am not sure if a question of such a type is suitable here.