1. ## Convergent series

Show that if $\displaystyle a_n \geq 0$ and $\displaystyle \sum_{n=1}^{\infty}a_n<\infty}$, then we can find a $\displaystyle b_n$ sequence that $\displaystyle \frac{b_n}{a_n} \rightarrow \infty$, but $\displaystyle \sum_{n=1}^{\infty}b_n<\infty$ is true.
I.e for any convergent series, exists an asymptotically greater convergent series.

$\displaystyle \displaystyle a_n \sim \frac{1}{n^{1+q}}$
$\displaystyle \displaystyle b_n \sim \frac{1}{n^{1+q/2}}$