Abel's Test: Let $f_n(x)$ be a non-increasing sequence of functions such that $0 \le f_n(x) \le M$ for all $x \in [a,b]$. If $\sum a_n$ converges then $\sum a_nf_n(x)$ converges uniformly in $[a,b]$.
Abel's Test corollary: Let $f_n(x)$ be a monotonic functions such that $0 \le f_n(x) \le M$ for all $x \in [a,b]$. If $\sum a_n$ converges then $\sum a_nf_n(x)$ converges uniformly in $[a,b]$.
my question is The way corollary get rid of monotonic decreasing positive condition and makes it easier by just having monotonic and uniformly bounded condition,
Can we do similarly in dirichlet's test and Dini's theorem?
ie monotonic decreasing condition of dirichlet's test and monotonic increasing condition of dini's thereom be modified to only monotonic condition?