Comparision test examples for $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

Question

Need some good examples that converges and some that diverges.

If $a_n>0, b_n>0$ for n = 1, 2, 3, . . . and {$a_n/b_n$}, {$b_n/a_n$} are both bounded sequences then $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ either both converge or both diverge.

Thanks

Answer 1

$\{n\}$ and $\{n+1\}$

$\{\dfrac{1}{n}\}$ and $\{\dfrac{1}{n+1}\}$

Answer 2

An example with both convergent series is just a variation of @MyGlasses's answer: $$ a_n=\frac{1}{n^2} \qquad b_n=\frac{1}{n^2+1} $$ It's interesting to note that the hypothesis of positiveness is necessary, otherwise conditionally convergent series provides counterexamples, for instance we could take $a_n=\frac{(-1)^n}{n}$ and $b_n=|a_n|$.