An example of series such that $0\le a_n\le b_n$ and $\sum a_n$ converges but $\sum b_n$ diverges

Question

Looking to get some help with the following question.

I need to find nonnegative sequences $(a_n)$and $(b_n)$ such that $a_n\leq b_n$ for all $n$ and $\sum a_n$ converges but $\sum b_n$ diverges.

So the terms of $a_n$ must turn to $0$ but $b_n$ does not. But how do I find examples of this series?

Answer 1

Well, if $a_n=\frac1{2^n}$ and $b_n=n$, then

$$a_n\le b_n$$

$$\sum_{n=1}^\infty\frac1{2^n}=1<+\infty$$

$$\sum_{n=1}^\infty n=+\infty$$

But it is not necessary for $\require{cancel}b_n\cancel\to0$, for example, with $b_n=\frac1n$,

$$\sum_{n=1}^\infty\frac1n>\int_1^\infty\frac1x\ dx=\lim_{t\to\infty}\ln(t)=+\infty$$

But it is necessary for $a_n\to0$ by the term test.