I know both Abel's and Dirichlet's test for convergence of some integrals when the range of the integral is of the form $[a,\infty)$, $a > 0$ (the statement of Abel's test canbe found in Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts). Is there any similar test for convergence of integrals by the name of Abel and Dirichlet when the range is a bounded interval of the form $[a,b]$, $0 < a < b$? i.e. want to conclude about the integrals of the form $\int_a^{b} f(x) * g(x)dx$.
Any reference or any kind of further help is highly appreciated. Thanks.
