Are there ways to test conditional convergence of $\sum_{n \geq 0} a_n$ (of course after the failure of total, absolute and uniform convergence) when Leibniz rule cannot be used and the series is not telescopic nor it can be rewritten as a known series (like the geometric or the harmonic ones), but the necessary condition for convergence $$\mathrm{lim}_{n \to \infty} a_n=0$$is still valid?
Suppose for example that the series is such that the general terms are not of constant sign, neither with alternating sign but still $\mathrm{lim}_{n \to \infty} a_n=0$.
What are the convergence tests used in such situations (if there are any)?
For problems like
$$\sum_{n=1}^\infty{\cos(n)\over n}$$
Dirichlet test is easily applicable. It is much more general than the alternating series test, though it holds the same basic idea.
Summations of the form $\sum a_nb_n$ converge if
$b_n$ is positive and monotonically decreasing.
$b_n\to0$ as $n\to\infty$
There exists an $M$ such that $\left|\sum_{n=a}^ba_n\right|<M$ for all $a<b$ in the summation.