Problem: Two monotonic series $\{a_n\}$ and $\{b_n\}$, $a_n\geq0, b_n\geq 0$ ,$\forall n \in \mathbb{N}^*$.Both $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are divergent, but $\sum_{n=1}^{\infty}\min\{a_n,b_n\}$ is convergent. Construct the sreies $\{a_n\}$ and $\{b_n\}$.
My Thoughts:
$\{a_n\}$ and $\{b_n\}$ are monotonic, and $\sum_{n=1}^{\infty}\min\{a_n,b_n\}$ is convergent, so $a_n\to 0,b_n \to 0(n \to +\infty)$.
$\sum_{n=1}^{\infty}\min\{a_n,b_n\}$ is convergent, so $\min\{a_n,b_n\}\to 0 (n \to +\infty)$.
Both $\sum_{n=1}^{\infty}a_n$ and $\sum_{n=1}^{\infty}b_n$ are divergent, so we can't determine whether $a_n\geq b_n$ or $b_n \geq a_n$ when $n$ is sufficiently large. So maybe $a_n$ and $b_n$ are trigonometric functions or mixtures of different series . ($\frac {1}{n}$ and $\frac{1}{n^2}$ for example)
Please give me some hints, thanks a lot!
