I have spent almost 2 hrs on this question, but I couldn’t find any good example for this one. The question is: Find sequence $(a_n)$ and $(b_n)$ satisifying: a) $a_n>0$, and $a_n \rightarrow0$; b) The partial sum of $b_n$ is a bounded sequence; and, c) $\sum a_nb_n$ diverges.
I felt like if this problem is somehow impossible, as for me, the conditions are exactly that of Abel’s test except that $a_n$ need not be decreasing. I need help on this one....
Take$$a_n=\begin{cases}\frac1n&\text{ if $n$ is odd}\\\frac1{n^2}&\text{ otherwise}\end{cases}$$and $b_n=(-1)^n$.