Can Abel's test and Dirichlet test be used interchangably? I have seen in most cases in series of constants and in series of functions that where Abel's test of convergence applies Dirichlet's test also applies and vice versa. This makes me to think whether the two tests are equivalent in the sense that  one is derivable from other.Is it true? Or there are cases where we can apply one but cannot apply other. Also is it so that one is more general and other is a corollary out of it.

The proofs have some commonality but one test is not a corollary of the other.
Suppose the sequence $(a_n)$ is not monotone and $\sum a_n$ converges. It follows that $\sum(1+1/n)^n a_n$ converges by Abel’s test. However, the conditions for Dirichlet’s test are not met since $(1+1/n)^n$ is not monotone decreasing — although it is bounded and monotone increasing.
