Is the converse of Dirichlet convergence test true?

Question

Dirichlet convergence test for series says:

Condition 1: Let $\{ a_n \}$ is a monotonically decreasing sequence of real numbers

Condition 2: Let $\{ b_n \}$ is a sequence of real numbers such that for all $N,$ $ \sum_{n=1}^N b_n < M $ where is $ M $ is a constant (not dependent on $N$

Example: $ \sum_{n=1}^\infty \frac{1}{n} \sin n $ is convergent.

If conditions 1 and 2 are true, then $ S = \sum_{n=1}^\infty a_n b_n $ is convergent.

My question is: is the converse also true? That is, if $ S $ is convergent and condition 1 is true, then does it imply condition 2 is also true? Or if $S$ is convergent and condition 2 is true then does condition 1 follow?

Answer 1

No, to both.

For the first: take $$a_n=\frac 1{n^3}\quad b_n=n$$ then $\sum a_nb_n$ converges, condition $1$ is met, but condition $2$ fails.

For the second: swap $a_n, b_n$ in the above.